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THE  ANALYTICAL  EXPRESSION  OF  THE  RESULTS 
OF  THE  THEORY  OF  SPACE-GROUPS 


-*.--'iar- 


BT 

RALPH  W.  G.  WYCKOFF 


PUBUSHED  BY  THE  CARNEGIE  INSTITUTION  OF  WASHING^N 

Washington,  October,  1922 


V'^ 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
Publication  No.  318 


THE  TECHNICAL  PRESS 
WASHINGTON,  D.  C. 


PREFACE. 

With  the  development  of  such  methods  of  stud5dng  the  arrangement  of 
the  atoms  in  crystals  as  are  furnished  by  the  phenomena  of  the  diffraction 
of  X-rays,  the  geometrical  theory  of  space  groups  becomes  of  the  utmost 
importance.  Until  recently  the  work  published  upon  this  theory  has  been 
directed  primarily  toward  the  preparation  of  a  statement  of  all  the  different 
kinds  of  symmetry  (groupings  of  elements  of  symmetry)  which  are  crystal- 
lographically  possible. 

This  statement,  to  be  complete,  must  necessarily  give  all  of  the  possible 
ways  of  arranging  points  in  space  which,  by  their  arrangement,  will  express 
crystallographic  symmetry.  In  its  most  general  form  such  an  analytical 
expression  of  the  results  of  this  theory  was  given  by  Schoenflies.*  Before 
it  is  applicable  to  the  study  of  the  structures  of  crystals,  however,  modifica- 
tions in  this  original  representation  are  necessary.  First,  there  must  be 
selected  such  a  portion  of  the  grouping  that  in  its  calculated  effects  upon 
X-rays  it  can  be  taken  as  typical  of  the  entire  arrangement.  It  is  thus 
necessary  to  state  a  space  group  in  terms  of  the  equivalent  positions  which 
lie  within  a  unit  cell  rather  than  by  giving,  as  Schoenflies  does,  the  equiva- 
lent positions  about  one  point  of  the  lattice  underlying  the  grouping.  This 
rather  obvious  modification  has  of  course  been  made  by  those  who  have 
used  the  space  groups  as  a  guide  in  studying  crystals.  The  second  modifi- 
cation, or  rather  amplification,  is  not  so  readily  made.  The  X-ray  experi- 
ments which  have  already  been  carried  out  show  that  the  number  of  particles 
(atoms)  contained  in  the  unit  cell  is  commonly  smaller  than  the  number  of 
most  generally  placed  equivalent  points  of  the  space  group  having  the  sym- 
metry of  the  crystal.  The  special  arrangements  of  the  equivalent  points 
(upon  axes,  planes,  and  other  elements  of  symmetry),  whereby  the  number 
of  most  generally  placed  equivalent  positions  is  reduced,  are  thus  of  great 
importance  and  it  becomes  essential  to  be  able  to  state  all  of  them  in  any 
particular  case.  Nigglif  has  already  given  the  simpler  of  these  special  cases. 
For  some  time  the  writer  has  been  engaged  in  working  out  all  of  them  and 
the  following  tables  are  an  expression  of  the  results  of  these  computations. 

It  was  the  original  intention  simply  to  state  these  results  and  to  outline 
the  method  whereby  they  were  obtained.  The  writer  is  firmly  convinced, 
however,  that  sure  and  definite  progress  in  this  relatively  new  field  of  crystal 
structure  can  be  realized  only  by  making  the  fullest  possible  use  of  the  added 
information  which  the  theory  of  space-groups  furnishes;  and  since  any  dis- 
cussion of  this  theory  is  almost  completely  absent  from  work  published  in 
English,  it  has  seemed  worth  while  to  add  a  brief  introduction  in  order  to 
give  such  details  of  the  space  groups  and  of  their  development  as  seem  to 
furnish  sufficient  background  for  the  appreciation  of  the  importance  of  the 
theory  in  this  new  field  of  physical  science. 

*  Krystallsysteme  und  Erystallstructur  (Leipzig,  1891). 

t  Geometrische  Krystallographie  des  Discontinuums  (Leipzig,  1919). 

Ill 

3005  25 


IV  PREFACE. 

At  present  a  Imowledge  of  the  method  of  derivation  is  not  required  by 
the  crystal  analyst  or  by  the  person  primarily  interested  in  using  the  results 
of  such  X-ray  studies.  Those  interested  in  the  theory  as  a  geometrical 
problem  will  of  course  find  the  development  thoroughly  given  by  Schoenflies. 
In  the  only  publication  available  to  Enghsh  readers  Hilton*  has  summarized, 
in  excellent  form,  the  work  of  Schoenflies,  introducing  at  the  same  time  some 
of  the  methods  of  representation  employed  by  Federov.  A  thorough  under- 
standing of  the  manner  of  developing  the  theory,  however,  is  best  attained 
from  a  study  of  the  original  work  of  Schoenflies.  The  discussion  in  the  pres- 
ent book  is  intended  for  those  who  wish  only  to  get  a  sufficient  idea  of  the 
nature  of  the  results  of  the  theory  of  space-groups  so  that  these  results 
can  be  intelligently  used. 

For  the  substance  of  this  discussion  the  writer's  obligation  to  Schoenflies 
is  obvious;  the  work  of  Hilton  has  also  been  used  with  entire  freedom.  In 
the  book  to  which  reference  has  already  been  made  Niggli  has  given  the 
positions  within  the  unit  cell  (of  each  space-group)  of  all  of  its  elements  of 
symmetry.  This  information,  while  of  no  aid  in  the  actual  determination 
of  the  structures  of  crystals,  may  prove  useful  in  the  attempt  to  derive  from 
these  structures  additional  information,  such  as  that  bearing  upon  the  internal 
symmetries  of  their  constituent  atoms.  In  comparing  the  partial  analytical 
expression  given  by  Niggli  with  his  results  based  directly  upon  those  of 
Schoenflies,  the  writer  found  that  particularly  in  the  case  of  the  tetragonal 
space-groups  there  were  many  differences,  owing  chiefly  to  the  choice  of 
different  points  as  the  origin  of  coordinates.  Because  of  the  possible 
usefulness  of  the  additional  data  relating  to  the  positions  of  elements  of 
symmetry  that  are  furnished  by  Niggli,  it  has  seemed  desirable,  in  spite  of 
some  loss  of  logicality,  to  recalculate  these  groups  so  that  they  would  accord 
with  those  already  published.  Similar  differences  exist  in  orthorhombic  and 
monoclinic  groups;  the  changes  necessary  to  reconcile  the  two  descriptions 
are  in  these  cases  sufficiently  obvious,  however,  that  it  has  seemed  worth 
while  only  to  indicate  in  some  more  or  less  illustrative  instances  the  nature 
of  the  translations  necessary  to  bring  about  a  general  coincidence. 

The  writer  wishes  to  express  his  gratitude  to  Dr.  S.  Nishikawa  for  the 
advice  and  criticism  given  him  when  in  1917  he  began  to  familiarize  himself 
with  the  theory  of  space-groups. 

Geophysical  Laboratory, 

March,  1921. 

*  Mathematical  Crystallography  (Oxford,  1903). 


CONTENTS. 

Paob 

Chapter  I.   Historical  Introduction 1-3 

Chapter  II.   Nature  of  the  Space-Groups 4-38 

Elements  of  Symmetry 4 

Point-Groups 6 

Analytical  Representation  of  the  Point-Groups: 

TricUnic  System 11 

Monochnic  System 12 

Orthorhombic  System 13 

Tetragonal  System 14 

Cubic  System 16 

Hexagonal  System 18 

Space  Lattices 22 

Space-Groups 24 

An  Outline  of  the  Derivation  of  the  Space-Groups: 

Triclinic  System 26 

Monochnic  System 27 

Orthorhombic  System 27 

Tetragonal  System 30 

Cubic  System 33 

Hexagonal  System 35 

Chapter  III.   The  Apphcation  of  the  Theory  of  Space-Groups  to  Crystals 39-46 

Units  of  Structure 39 

Space-Groups  and  Crystals 42 

Special  Cases  of  the  Space-Groups 44 

The  Treatment  of  Calcite  as  a  Typical  Case 45 

Chapter  IV.  The  Complete  Analytical  Expression  of  the  Space-Groups 47-180 

Trichnic  System: 

A.  Hemihedry,  Ci 48 

B.  Holohedry,  Cj 48 

Monochnic  System: 

A.  Hemihedry,  Cg 49 

B.  Hemimorphy,  C2 49 

C.  Holohedry,  C| 49 

Orthorhombic  System: 

A.  Hemimorphy,  C2 52 

B.  Hemihedry,  V 56 

C.  Holohedry,  V*' 59 

Tetragonal  System: 

A.  Tetartohedry  of  the  Second  Sort,  S4 73 

B.  Hemihedry  of  the  Second  Sort,  V*^ 73 

C.  Tetartohedry,  C4 79 

D.  Paramorphic  Hemihedry,  C4 80 

E.  Hemimorphic  Hemihedry,  C4 83 

F.  Enantiomorphic  Hemihedry,  D4 86 

G.  Holohedry,  D^ 89 

V 


V I  CONTENTS. 

Cubic  Sj'^stem: 

The  Special  Cases  of  the  Cubic  System 103 

A.  Tetartohedry,  T 121 

B.  Paramorphic  Hemihedry,  T*^ 123 

C.  Hemimorphic  Hemihedry,  T   128 

D.  Enantiomorphic  Hemihedry,  O 132 

E.  Holohedry,  O^ 138 

Hexagonal  System — Rhombohedral  Division: 

A.  Tetartohedry,  C3 151 

B.  Hexagonal  Tetartohedry  of  the  Second  Sort,  C3 151 

C.  Hemimorphic  Hemihedry,  C3 152 

D.  Enantiomorphic  Hemihedry,  D3 153 

E.  Holohedry,  D^ 155 

Hexagonal  System — Hexagonal  Division: 

A.  Trigonal  Paramorphic  Hemihedry,  C3 157 

B.  Hemihedry  with  a  Three-fold  Axis,  D'3* 158 

C.  Hexagonal  Tetartohedry,  Cq 160 

D.  Hemimorphic  Hemihedry,  Cg 161 

E.  Paramorphic  Hemihedry,  Cg 162 

F.  Enantiomorphic  Hemihedry,  Dg 163 

G.  Holohedry,  Dg 166 

Summarizing  Tables 170 

TABLES. 

Table  1.   A  Comparison  of  Some  Current  Systems  of  Point-Group  (Crystal  Class) 

Nomenclature 10 

Table  2.   The  Unit  Cells  of  Each  of  the  14  Space  Lattices 42 

Tables  Summarizing  the  Numbers  op  Special  Cases  op  Space-Groups  Having 
THE  Symmetry  op  Each  op  the  Systems  op  Crystal  Symmetry: 

Table  3.   Triclinic  System 170 

Table  4.   Monochnic  System 170 

Table  5.   Orthorhombic  System 171 

Table  6.   Tetragonal  System 174 

Table  7.   Cubic  System 176 

Table  8.   Hexagonal  System — Rhombohedral  Division 178 

Table  9.   Hexagonal  System — Hexagonal  Division 179 


THE  ANALYTICAL  EXPRESSION  OF 

THE  RESULTS  OF  THE  THEORY 

OF  SPACE-GROUPS 


By  Ralph  W.  G.  Wyckoff 


vn 


CHAPTER  I. 
HISTORICAL  INTRODUCTION.* 

The  investigation  of  the  structure  of  crystals  involves  the  study  both  of 
the  substance  from  which  the  crystals  are  made  and  of  the  way  in  which  this 
material  is  arranged  in  space.  Until  very  recently,  practically  all  of  the 
information  bearing  upon  the  first  of  these  points  has  arisen  from  the  realiza- 
tion of  the  probable  physical  reality  of  the  chemical  atom.  How  these 
atoms  are  associated  together  in  crystals  and  whether  the  chemical  mole- 
cule, or  some  other  aggregate  of  atoms,  has  the  significance  in  solids  which 
it  possesses  in  gases  and  Hquids  are  questions  which  have  been  answered 
only  by  conjecture  and  inference.  The  development  in  the  other  direction, 
however,  presenting  a  problem  which  in  its  most  general  statement  is  inde- 
pendent of  current  hypotheses  concerning  the  nature  of  the  material  from 
which  crystals  are  built,  has  been  capable  on  the  other  hand  of  a  far-reaching 
and  apparently  satisfactory  growth. 

In  the  days  when  an  atomic  structure  of  matter  was  a  crude  working 
hypothesis  without  any  basis  in  experimentally  determined  fact,  we  find 
Robert  Hookef  reproducing  the  forms  of  alum  by  properly  pihng  up  "a 
company  of  bullets  and  some  few  other  very  simple  bodies,"  very  much  as 
we  represent  the  structure  of  a  crystal  on  the  basis  of  X-ray  measurements. 

It  was  the  phenomenon  of  regular  cleavage,  however,  that  suppHed  the 
evidence  upon  which  early  hypotheses  of  the  regular  arrangement  of  the 
material  of  crystals  were  based.  For  instance,  Westfeld|  considered  calcite 
as  built  up  of  tiny  rhombohedrons;  and  Bergman,§  basing  his  behefs  partly 
on  the  observation  of  Gahn  that  a  skalenohedron  of  calcite  yields  a  rhombo- 
hedron  on  cleaving,  developed  what  might  be  called  the  first  geometrical 
theory  of  crystal  structure.  For  just  as  the  crystals  of  calcite  could  be 
considered  as  an  aggregate  of  minute  rhombohedrons  placed  parallel  to  one 
another,  so  garnet  or  pyrite  or  other  crystals  can  be  developed  similarly  from 
certain  fundamental  forms.  These  ideas  seem  to  be  essentially  the  same 
as  those  held  by  Hauy.^  He,  also,  considered  cleavage  as  the  guiding 
factor.  The  cleavage  units,  his  molecules  iniegrantes,  were  either  tetrahedra, 
triangular  prisms,  or  parallelopipeda,  and  he  showed  how  crystals  with  vari- 
ously developed  faces  could  be  represented  by  the  aggregation  of  these  units. 
These  ideas  of  Hauy  were  built  around  the  law  of  rational  indices,  though 
they  were  fundamentally  independent  of  it.  Many  objections  to  the  details 
of  the  hypothesis  of  Hauy  arose,  as  indeed  they  must  arise  against  any  theory 
based  primarily  upon  cleavage.     Not  only  does  the  existence  of  the  many 

*  Most  of  the  material  for  this  introduction  is  given  by  L.  Sohncke,  Entwickelung  einer 
Theorie  der  Krystallstruktur  (Leipzig,  1879).  It  is  given  in  English  and  brought  up  to  date  in  a 
report  of  the  Brit.  Assoc.  297-337.     1901. 

t  Micrographia  (London,  1665),  p.  85. 

t  Mineralogische  Abhandlungen,  Stuck  I.     1767. 

§  Nov.  Acta.  Reg.  Soc  Se.  Upsal.  1773,  i;  Opusc.  (Upsala)  1780,  ii. 

q  Essai  de  Cristallographie  (Paris)  1772;  etc. 


r^;.^". 


HISTORICAL   INTRODUCTION. 


crystals  which  show  no  cleavage  necessitate  many  supplementary  hypotheses, 
but  the  observed  cleavage  of  such  substances  as  fluorite  (with  octahedral 
cleavage)  is  not  readily  accounted  for  by  any  kind  of  close-fitting  units. 

Simultaneously  with  the  extension  of  the  belief  in  the  atomic  nature  of 
substances,  and  perhaps  because  of  this  belief,  emphasis  came  to  be  shifted 
from  the  shape  of  the  crystal  units  to  the  relative  positions  of  their  centers 
of  gravity  as  centers  of  some  sort  of  crystal  molecules.  Thus  there  evolved 
from  these  different  speculations  the  basis  for  a  suitable  geometrical  study 
in  the  definite  conception  of  a  crystal  as  composed  of  units  of  undefined  shape 
repeated  in  some  regular  fashion  throughout  space. 

In  such  a  regular  pattern  for  repeating  the  crystal  unit  we  have  a  space 
lattice.  All  of  the  symmetrical  networks  of  points  which  can  have  crystallo- 
graphic  symmetry  were  found  geometrically  by  Frankenheim.  *  Some  years 
later  this  was  done  more  accurately  and  rigidly  by  Bravais.f  As  a  result  of 
his  work,  Bravais  looked  upon  a  crystal  as  built  up  by  placing  units  of  a 
suitable  symmetry  all  in  the  same  orientation  at  the  points  of  one  of  these 
symmetrical  networks.  Thus  the  unit  of  a  cubic  crystal  might  have  cubic 
or  even  tetrahedral  symmetry,  but  it  could  not,  for  instance,  have  monoclinic 
or  hexagonal  symmetry.  As  a  matter  of  fact,  Bravais  thought  of  his  units 
as  groups  of  atoms  forming  some  sort  of  a  crystal  molecule,  though  such  a 
view  is  not  a  necessary  part  of  the  geometrical  development.  In  this  theory 
of  Bravais,  in  which  a  crystal  is  composed  of  aggregates  of  atoms  repeated 
regularly  and  indefinitely  through  space,  is  to  be  found  the  beginning  of  an 
adequate  treatment  of  the  possible  groupings  of  matter  in  crystalline  bodies. 
The  objections  to  Bravais'  theory,  however,  are  many  and  obvious.  In  the 
first  place,  all  of  the  space  lattices  have  the  complete  symmetry  of  some  one 
of  the  crystal  systems,  so  that,  in  order  to  account  for  the  lower  degrees  of 
symmetry,  it  was  necessary  for  him  to  ascribe  the  degradation  in  such  cases 
to  the  shape  of  the  crystal  units,  or  molecules,  without  at  the  same  time 
being  able  satisfactorily  to  treat  these  units.  Again  this  theory  implies  a 
distinct  restriction,  and  one  which  had  not  been  proved  necessary,  that  all 
of  the  crystal  molecules  must  have  the  same  orientation  throughout  the 
crystal. 

In  the  course  of  a  general  study  of  the  theory  of  groups  of  movements 
JordanI  gave  a  perfectly  general  method  for  defining  all  of  the  possible  ways 
of  regularly  repeating  an  identical  grouping  of  points  indefinitely  throughout 
space.  By  combining  this  treatment  of  Jordan  with  the  principle  (laid 
down  by  Wiener)  that  regularity  in  the  arrangement  of  indentical  atoms  is 
attained  when  "every  atom  has  the  other  atoms  arranged  about  it  in  the 
same  fashion,"  Sohncke§  eventually  deduced  all  of  the  typical  ways  of  regu- 
larly repeating  identical  groupings  of  atoms  throughout  space  so  that  the 

*  Die  Lehre  von  der  Cohasion  (Breslau,  1835). 

t  Journ.  de  I'ficole  Polytech.  (Paris)  XIX,  127.     1850;  XX,  102.     1851. 
t  Annali  di  matematica  pura  ed  applicata  (2)  2,  167,  215,  322.     1869. 
§  L.  Sohncke,  op.  cit. 


HISTOEICAL   INTRODUCTION.  3 

total  assemblage  will  possess  erystallographic  symmetry.*  This  method  of 
treatment  in  attacking  the  problem  of  the  arrangement  of  the  points  within 
what  was  the  crystal  unit  or  molecule  of  Bravais  brings  the  problem  towards 
its  final  solution. 

None  of  the  systems  of  Sohncke  can  be  made  to  account  in  an  entirely 
satisfactory  manner  for  the  enantiomorphic  (mirror-image)  characteristics  of 
many  crystals.  SchoenfUesf  was  led  to  consider  that  every  point  of  an 
assemblage  must  have  all  of  the  other  points  ranged  about  it  in  a  "hke  fash- 
ion," where  "likeness"  may  refer  either  to  an  identical  arrangement  or  to 
a  mirror-image  similarity.  Starting  from  this  basis,  he  obtained  the  230 
space  groups  which  represent  all  of  the  possible  typical  ways  of  arranging 
(symmetry-less)  points  in  space  so  that  the  grouping  will  possess  the  sym- 
metry of  one  of  the  thirty-two  crystal  classes.  The  same  derivation  of  the 
space  groups  was  accomplished  independently  by  Federov|  and  by  Barlow, 
but  at  present  the  work  of  Schoenflies  is  the  most  useful  because  it  is  pre- 
sented in  a  form  that  is  of  immediate  application.  With  the  aid  of  this 
final  theory  of  space  groups  the  different  degrees  of  symmetry  exhibited  by 
crystals  can  at  last  be  traced  back  definitely  and  precisely  to  the  arrangement 
of  the  atoms  in  the  crystals  (without  postulating  any  characteristics  of  sym- 
metry for  them) . 

Besides  indicating  the  elements  of  symmetry  which  are  characteristic  of 
each  of  the  230  typical  ways  of  arranging  points  in  space,  Schoenflies  gives, 
in  general  terms,  the  coordinates  of  the  points  in  each  of  these  groupings 
which  are  equivalent  to  one  another. 

The  discovery  of  the  diffraction  of  X-rays  and  the  consequent  develop- 
ment of  the  physical  methods  for  studying  the  structure  of  crystals  have  made 
this  analytical  expression  of  the  results  of  the  theory  of  space  groups  of  the 
utmost  importance.  It  is  the  purpose  of  the  present  work  to  give  these 
results  a  detailed  expression,  thereby  putting  them  into  a  form  in  which  they 
will  be  immediately  useful  as  an  aid  to  the  study  of  the  arrangement  of  the 
atoms  in  crystals.  X-ray  experimentation  thus  far  carried  out  shows  that 
the  special  cases  which  result  when  equivalent  points  (the  atoms  in  crystals) 
lie  in  some  element  or  elements  of  symmetry,  such  as  axes  or  planes,  are  the 
ones  which  are  physically  most  important.  As  a  consequence  the  prepara- 
tion of  this  detailed  expression,  in  so  far  as  it  introduces  material  which  is 
not  outlined  in  the  work  of  Schoenflies,  has  made  necessary  the  working  out 
of  all  of  these  special  cases  for  all  of  the  space-groups. 

*  At  first  Sohncke  seems  to  have  been  inclined  to  view  all  of  the  points  of  a  point  system  as 
regular  and  all  of  one  kind.  When  the  insufficiency  of  this  theory  was  emphasized  he  postulated 
the  presence  of  a  few  different  kinds  of  points  (which  can  be  made  to  correspond  with  different 
kinds  of  atoms).  The  partial  grouping  composed  of  the  points  of  any  one  kind  is  homogeneous; 
at  the  same  time  the  different  groupings  all  have  the  axes  and  the  other  elements  of  symmetry 
in  common. 

t  A.  Schoenflies.  Krystallsysteme  u.  Krystallstruktur  (Leipzig,  1891). 

t  E.  Federov.  Z.  Kryst.  24,  209.  1895;  W.  Barlow.  Z.  Kryst.  23,  1.  1894.  Federov's  work 
appeared,  in  Russian,  before  that  of  either  of  the  other  two. 


CHAPTER  II. 
NATURE  OF  THE  SPACE  -  GROUPS. 

ELEMENTS   OF  SYMMETRY. 

Axes  of  symmetry. — An  axis  of  rotation  of  a  figure*  is  a  line  about  which 
the  figure  can  be  rigidly  turned.  The  angle  of  the  rotation  is  the  angle 
between  the  final  and  initial  positions  of  a  plane  which  contains  the  axis 
of  rotation.  A  figure  is  said  to  possess  an  axis  of  symmetry  when  rotation 
through  a  definite  angle  about  an  axis  of  rotation  will  cause  the  figure  to 
assume  the  same  point-for-point  configuration  that  it  originally  possessed. 
The  angle  of  the  rotation  about  an  axis  which  is  required  to  bring  about 
this  coincidence  is  called  the  angle  of  the  axis  of  symmetry.  Every  figure 
has  an  infinite  number  of  27r  axes  of  symmetry;  that  is,  a  complete  rotation 
of  360"  about  any  line  through  a  body  will  cause  it  to  assume  its  original 
configuration.  The  operation  of  such  a  27r  (one-fold)  axis  is  called  the  iden- 
tical operation  of  symmetry  (or  simply  the  identity).  If  a  rotation  of  180° 
is  sufficient  to  effect  a  coincidence,  the  axis  of  rotation  is  a  180",  or  two-fold 
axis  of  symmetry;  more  generally,  an  n-fold  axis  of  symmetry  is  one  for  which 

27r 
a  rotation  of  angle  —  brings  about  coincidence.     One-,  two-,  three-,  four-  and 
n 

six-fold  axes  are  found  in  crystals  (and  in  figures  possessing  crystallographic 

symmetry).     (Figure  1.) 


\ 


\ 


I 


r 

60\ 


/ 

/I 


/ 


>a*^ 


Fig.  1.  The  crystallographically  significant  rotational  axes  of  symmetry. 

PZane  of  symmetry. — In  figure  2  the  line  POP'  is  perpet  dicular  to  the  plane 
ABCD.  If  then  PO  equals  OP'  in  length,  the  point  P'  stands  in  a  mirror- 
image  relation  to  the  point  P.  If  a  plane  can  le  passed  through  a  figure  so 
that  every  point  of  the  figure  upon  one  side  of  this  plane  has  a  corresponding 


*  By  a  figure  is  meant  any  sort  of  a  collection  of  points,  lines,  planes,  and 

4 


so  on. 


ELEMENTS  OF  SYMMETRY.  5 

point  in  a  mirror-image  position  upon  the  other  side  of  the  plane,  the  plane 
is  a  plane  of  symmetry. 

Center  of  sijmmeiry. — A  point  of  a  figure  is  a  center  of  symmetry  if  a  line 
drawn  from  any  point  of  the  figure  to  it  and  extended  an  equal  distance 
beyond  will  encounter  a  point  corresponding  to  the  arbitrarily  chosen  point. 
(Figure  3.) 


Fig.  2. 


Fig.  3.  O  is  the  center  of  symmetry  of  a  figure  in 
which  P  and  Pi  are  corresponding  points. 


Screw-axes  of  symmetry. — A  figure  is  said  to  experience  a  translation  when 
every  point  of  the  figure  is  moved  by  the  same  amount  in  the  same  direction. 
A  rotation  about  an  axis  accompanied  by  a  translation  along  the  axis  of 
rotation  is  called,  a  rotary  translation.      This  screw-motion  must  be  defined 


6 


THE   NATURE   OF  THE   POINT-GROUPS. 


both  by  the  angle  of  the  rotation  and  by  the  amount  of  the  translation. 
The  axis  of  the  rotation  (and  the  line  of  the  translation)  is  called  a  screw- 
axis.     If  such  a  rotary  translation  will  bring  the  points  of  a  figure  into  co- 


a- 


:> 


Fig.  4.  The  crystallographically  significant  screw  axes  of  symmetry. 

incidence,  the  axis  of  the  motion  is  a  screw-axis  of  symmetry.  In  a  figure 
having  crystallographic  symmetry  these  screw-axes  may  be  one-,  two-,  three-, 
four-  or  six-fold.     (Figure  4.) 

Glide  planes  of  symmetry. — If  a  figure 
can  be  brought  into  point-for-point  co- 
incidence by  a  reflection  in  a  plane 
combined  with  a"  translation  of  a  defi- 
nite length  and  direction  in  the  plane, 
the  plane  is  called  a  glide  plane  of 
symmetry.  In  this  case  the  transla- 
tion-reflection must  be  defined  both  by 
the  position  of  the  plane  and  by  the 
length  and  direction  of  the  translation. 
(Figure  5.) 

POINT-GROUPS. 

The  thirty-two  ways  of  suitably  com- 
bining these  planes,  axes,  and  centers 
of  sjnnmetry  give  the  elements  of  sym- 
metry which  are  characteristic  of  the 
32  classes  of  crystallographic  symme- 
try. Each  one  of  these  combinations  of 
symmetry  elements  is  a  point-group.  Thus,  a  point-group  may  be  defined  by 
stating  either  the  elements  or  operations*  of  symmetry  which  characterize  it. 

*  By  an  operation  of  sjonmetry  is  meant  any  movement  which  will  bring  about  a  point-for- 
point  coincidence.     For  instance,  a  six-fold  axis  of  rotation  possesses  six  operations  of  symmetry 


Fig.  5.  Pi  is  a  glide  reflection  of  P  in  the 
plane  shown  in  the  figure. 


THE   NATURE   OF   THE   POINT-GROUPS.  7 

The  elements  of  symmetry  characteristic  of  each  of  the  32  point-groups 
will  now  be  given. 

The  cycUc  groups  have  only  one  axis  of  symmetry.  They  may  be  written 
symbolically  as 

where  n  may  be  either  1,  2,  3,  4  or  6.  A  will  be  taken  as  the  symbol  of  a 
rotation  so  that  the  term  within  the  braces  is  to  be  considered  as  defining  a 

rotation  of  angle  — . 
n 

27r 
Dieder-groups  have  one  principal  axis  of  symmetry  of  angle  —  and  n  two- 
fold axes  in  a  plane  at  right  angles  to  the  principal  axis. 

where  U  (Umklappung)  will  be  used  to  represent  the  two-fold  rotation  of 
the  secondary  axes.  The  value  of  n  may  be  1,  2,  3,  4  or  6.  The  group  Di 
is  clearly  identical  with  C2,  however.  The  positions  of  the  axes  of  the  other 
groups  are  shown  in  figure  6.  The  group  for  which  n  =  2  furnishes  the  spe- 
cial case  of  three  two-fold  axes  at  right  angles  to  one  another;  this  group  is 
more  commonly  known  as  the  vierer-group  and  is  designated  as  V. 


Fig.  6. 

The  tetrahedral  group  (symbol  =  T)  has  3  two-fold  axes  at  right  angles 
to  one  another  (like  the  vierer-group)  and  4  three-fold  axes  so  placed  that  if 
the  two-fold  axes  are  taken  to  bisect  the  sides  of  a  circumscribed  tetrahedron, 
the  4  three-fold  axes  will  each  one  pass  through  the  point  of  intersection  of 
the  two-fold  axes  and  through  one  of  the  corners  of  the  tetrahedron  (figure  7). 

The  octahedral  group  (symbol  =  0)  has  3  four-fold,  4  three-fold,  and  6 
two-fold  axes  arranged  in  the  same  manner  as  are  the  altitudes,  the  body- 
diagonals,  and  the  face-diagonals  of  a  cube  (figure  8). 

The  groups  which  have  so  far  been  considered  require  only  simple  rota- 
tion axes  for  their  expression;  they  are  commonly  called  groups  of  the  first 
sort.     Those  that  now  follow  are  groups  of  the  second  sort. 


8  THE   NATURE   OF   THE   POINT-GROUPS. 

Cyclicgroups  of  the  second  sort  possess  one  screw-axis  of  symmetry 

=■  =  {<!)}■ 

where  the  symbol  in  brackets  may  be  taken  as  a  rotary  translation  of  angle 

— .    The  value  of  n  may  be  1,  2,  3,  4  or  6.     When  n  =  1  the  rotary  translation 

is  clearly  equivalent  to  a  reflection  in  a  plane  at  right  angles  to  the  axis  of 
rotation.  Thus,  in  figure  9a  the  rotary  translation  of  angle  27r  will  bring 
the  point  P  to  the  position  Pi;  this  operation  is,  however,  equivalent  to  a 


reflection  in  the  plane  through  0  normal  to  pp',  where  Op  is  equal  to  one- 
half  of  the  length  r  of  the  translation  component  of  the  rotary  translation. 
When  n  =  2,  the  resulting  rotary  translation  Op  is  equivalent  to  an  inversion 
through  the  point  0  of  figure  9b.     These  two  groups  may  thus  be  written 

Ci={A(27r))  =  {S}     and     C2=  {A(7r)}  =  {1} 

where  S  (Spiegelung)  stands  for  a  reflection  and  I  for  an  inversion.  In  a 
similar  fashion  it  will  be  seen  that  when  n==4,  this  group  is  identical  with 

one  obtained  by  combining  a  rotation  A(  2  )  with  a  reflection  S^  in  a  hori- 
zontal plane  of  symmetry.     Thus 


C.  =  Ki)}  =  {A(i>Bfs.. 


Other  groups  of  the  second  sort  can  be  obtained  by  combining  a  principal 
axis  of  rotation  with  a  plane  or  with  a  center  of  symmetry.  Three  types  of 
such  groups  having  but  one  axis  of  symmetry  are  possible:  (1)  when  the  plane 
of  symmetry  is  normal  to  the  axis  of  symmetry  (a  horizontal  reflecting  plane), 
(2)  when  the  plane  of  symmetry  contains  the  axis  of  symmetry  (a  vertical 


THE   NATURE   OF   THE   POINT-GROUPS. 


9 


reflecting  plane)  and  (3)  when  the  new  element  of  symmetry  is  a  center  of 
symmetry.     These  three  types  may  then  be  written 

(1)    CS=  {Cn,  Sh}  (2)    Cl=  {Cn,  Sv}  (3)    C^=  {Cn,  1} 

It  can  be  shown  that  if  n  is  odd,  all  three  of  these  types  are  possible.  When, 
however,  n  is  even,  the  number  of  different  groups  for  any  value  of  n  is  but 
two.    The  groups  of  this  sort  that  are  thus  possible  are  the  following: 

When  n  =  l. — The  group  Ci  is  clearly  the  same  as  the  group  Cl;  further- 
more it  is  identical  with  the  group  Ci.  Similarly  the  group  Ci  is  identical 
with  the  group  Cg. 


Fig.  8. 

When  n  =  2, 4  or  6.— CS  =  CL,  so  that  groups  of  the  types  CS  =  CL  and  Cl  are 
possible. 

When  n  =  3. — Point-groups  of  all  three  types  are  possible. 

Some  new  groups  arise  by  combining  the  axes  of  a  group  of  the  type  Dn 
with  a  reflection  plane.  The  plane  of  symmetry  may  Ue  in  the  horizontal 
position  (normal  to  the  principal  axis  of  symmetry) ;  if  it  lies  in  the  vertical 
position  new  groups  will  be  obtained  only  when  the  plane  bisects  the  angle 
between  secondary  axes  (a  diagonal  plane).  It  can  furthermore  be  shown 
that  in  the  latter  case  groups  of  crystallographic  significance  will  be  obtained 
only  when  n  =  2  and  when  n=3.    Thus,  when  n  =  2,  3,  4,  or  6,  we  have  the 

new  groups 

D|-V^,        D5  =  V^        Dg,        DS, 

The  groups  T*  and  T**  arise  from  the  tetrahedral  group,  T,  by  combining 
the  axes  of  T  with  a  horizontal  and  with  a  diagonal-vertical  reflecting  plane, 
respectively.  One  new  group,  O,  can  be  produced  from  the  octahedral 
group  O. 

All  of  the  32  groups  have  now  been  defined.  On  the  basis  of  their  total 
symmetry  these  32  point-groups  can  be  placed  in  6  (or  7)  systems,  the  systems 
of  crystallographic  symmetry.*  This  classification  of  the  point-groups  is 
given  in  Table  1,  together  with  the  names  of  the  classes  of  crystal  symmetry 
(according  to  Schoenflies,  Dana,  and  Groth)  corresponding  to  each. 

*  A  basis  for  this  classification  will  become  evident  when  the  point-groups  are  discussed  sep- 
arately and  given  an  analytical  expression. 


10 


NOMENCLATURE   OF   THE   POINT-GROUPS. 
Table  1. 


Symbol. 

Class  of  symmetry. 

S) 

a 
o 

1 

o 

d 

of  symmetry  and  | 
of  equivalent         j 
points.                    1 

SCHOENFUBB. 

Dana. 

GftOTH. 

I. 

Triclinic  system: 

1.  Ci 

Hemihedry 

Asymmetric 

Asymmetric  pedial 

1 

2.  C,«Sj=Ci 

II. 

Holohedry 
Monoclinic  system. 

Normal 

Pinacoidal 

2 

3.  Ci-C^-Cg 

Hemihedry 

Clinohedral 

Domatio 

2 

4.  C« 

Hemimorphic  hemihedry 

Hemimorphic 

Monoclinic  sphenoidal 

2 

5.CJ 

III. 

Holohedry 
Orthorhombic  system. 

Normal 

Monoclinic  prismatic 

4 

6.0- 

Hemimorphic  hemihedry 

Hemimorphic 

Rhombic  pyramidal 

4 

7.  D8=V 

Enantiomorphic  hemihedry  Sphenoidal 

Rhombic  bispbenoidal 

4 

8.  dJ-V** 

Holohedry 

Normal 

Rhombic  bipyramidal 

8 

IV. 

Tetragonal  system. 

^      9.    S4=C4 

Tetartohedry  of  second  sort  Tetartohedral 

Tetragonal  bispbenoidal 

4 

MO.  v'^-dJ 

Hemihedry  of  second  sort 

Sphenoidal 

Tetragonal  scalenohedral 

8 

^    11.  C« 

Tetartohedry 

Pyramidal  hemimorphic 

Tetragonal  pyramidal 

4 

\  12.  Cj 

Paramorphic  hemihedry 

Pyramidal 

Tetragonal  bipyramidal 

8 

13.  Cj 

Hemimorphic  hemihedry 

Hemimorphic 

Ditetragonal  pyramidal 

8 

^  14.  D4 

Enantiomorphic  hemihedry  Trapezohedral 

Tetragonal  trapezohedral 

8 

\  16.  dJ 

V. 

Holohedry 
Cubic  system. 

Normal 

Diteti  agonal  bipyramidal 

16 

16.  T 

Tetartohedry 

Tetartohedral 

Tetrahedral  pentagonal  dode- 
cahedral 

12 

17.  T^ 

Paramorphic  hemihedry 

Pyritohedral 

Diacisdodecahedral 

24 

18.  T'^ 

Hemimorphic  hemihedry 

Tetrahedral 

Hexacistetrahedral 

24 

19.  0 

Enantiomorphic  hemihedry  Plagihedral 

Pentagonalicositetrahedral 

24 

20.  O'* 

VI. 

Holohedry 
Hexagonal  system. 

Normal 
Rhombohedral  Division 

Hexacisoctahedral 

48 

21.  C. 

Tetartohedry 

24. 

Trigonal  pyramidal 

3 

22.  C\ 

Hexagonal  tetartohedry  of 
second  sort 

Trirhombohedral 

Rhombohedral 

6 

23.  C^ 

Hemimorphic  hemihedry 

Ditrigonal  pyramidal 

Ditrigonal  pyramidal 

6 

24.  ^r^ 

25.  Df 

Enantiomorphic  hemihedry  Trapezohedral 

Trigonal  trapezohedral 

6 

Holohedry 

Rhombohedral 

Ditrigonal  scalenohedral 

12 

26.  Cj 

Hexaponal  Division 

Trigonal  paramorphic  hem 

-23. 

Trigonal  bipyramidal 

6 

_h 

hedry 

27.  D^ 

Trigonal  holohedry 

Trigonotype 

Ditrigonal  bipyramidal 

12 

28.  C« 

Tetartohedry 

Pyramidal  hemimorphic 

Hexagonal  pyramidal 

6 

29.  C, 

Paramorphic  hemihedry 

Pyramidal 

Hexagonal  bipyramidal 

12 

80.  C^ 

Hemimorphic  hemihedry 

Hemimorphic 

Dihexagonal  pyramidal 

12 

31.  D. 

Enantimorphio  hemihedry 

Trapezohedral 

Hexagonal  trapezohedral 

12 

82.  dJ 

Holohedry 

Normal 

Dihexagonal  bipyramidal 

24 

Note. — It  may  be  remarked  that  the  numbers  of  the  first  column  have  no  particular  sig- 
ni^p^ce  and  do  not  refer  to  any  of  the  current  systems  of  designating  symmetry  classes. 


THE   TRICLINIC   POINT-GROUPS. 


11 


The  analytical  expression  of  the  point-groups. — On  the  basis  of  the  defini- 
tions of  the  32  point-groups,  the  operations  of  symmetry  (footnote  on  page  6) 
that  characterize  each  of  them  can  be  immediately  written.  Furthermore, 
it  is  evident  that  if  any  point,  x,  y,  z,  is  subjected  to  each  of  the  operations  of 
a  point-group,  a  group  of  equivalent  points  will  result  whose  symmetry  is 


Fig.  9. 

that  of  the  point-group;  thus  its  analytical  expression  is  obtained.  The 
operations  of  S3anmetry  which  are  characteristic  of  each  of  the  point-groups 
will  now  be  stated  and  through  them  analytical  representations  given  to  each 

of  these  groups. 

TRICLINIC  SYSTEM. 

Point-group  Ci. — This  group  has  but  one  element  of  symmetry,  the  identical 
operation  (symbol  =1).  Since  the  identity  brings  any  point  x,  y,  z  into 
coincidence  with  itself,  any  single  point,  xyz,  serves  as  an  analytical  rep- 
resentation of  this  group.  The  coordinate  axes  to  which  these  coordinates 
refer  obviously  can  be  any  three  Unes  in  space,  of  unequal  unit  lengths  and 
making  unequal  angles  with  one  another.  Such  axes  will  be  called  the  tri- 
cUnic  axes  of  reference.  They  are  equally  serviceable  for  the  point-group, 
Ci,  which  follows. 

Pointr-group  d. — The  operations  of  S5rmmetry  characteristic  of  this  group 
are  the  identity  (obviously  an  operation  of  every  group)  and  an  inversion 


12 


THE   MONOCLINIC   POINT-GROUPS. 


(symbol  =  I).  Since  an  inversion  through  the  origin  of  coordinates  changes 
the  signs  of  all  three  coordinates  (figure  3)  the  operations  of  symmetry  and 
the  coordinates  of  equivalent  points  of  this  point-group  are 

Operations  of  symmetry:  1,  I. 

Coordinates  of  equivalent  points:  xyz;        xyz. 


MONOCLINIC  SYSTEM. 

In  their  analytical  expressions  all  of  the  point-groups  having  the  symmetry 
of  this  system  can  be  referred  to  a  system  of  axes,  two  of  which  (the  X-  and 
Y-axes)  make  any  angles  with  one  another;  the  third  axis  (the  Z-axis)  is  normal 
to  the  plane  of  these  other  two.  The  Z-axis  consequently  is  taken  to  coincide 
with  the  principal  two-fold  axis,  where  such  exists. 

Point-group  Cg. — The  single  operation  of  symmetry  (besides  the  identity) 
of  this  group  is  a  reflection  to  be  taken  in  the  horizontal  (XY-)  plane.     Since 
such  a  reflection  (symbol  =  Sii)  changes  the  sign  of  the  z-coordinate  (figure  10), 
the  operations  and  equivalent  and  equivalent  points  of  this  group  are 
Operations  of  symmetry:  1,  S^. 

Coordinates  of  equivalent  points:  xyz;        xyz. 


Fig.  10. 


Fig.  11. 


Point-group  C2. — A  two-fold  rotation  about  an  axis  (the  Z-axis)  normal 
to  the  plane  of  the  other  two  axes  of  coordinates,  changes  the  signs  of  these 
two  coordinates  (figure  11).  Consequently  the  point-group  C2  can  be  ex- 
pressed as 

Operations  of  symmetry:  1,  A(7r). 

Coordinates  of  equivalent  points:  xyz;        xyz. 

Point-group  C|.— Since  this  group  is  developed  by  mirroring  C2  in  a  hori- 
zontal (XY-)  plane  of  symmetry,  it  is  to  be  expressed  as  follows: 

Operations  of  symmetry:  1,        A(ir),        Sh,        A(7r)Sh. 
The  operation  whose  symbol  is  A(ir)Si„  the  product  of  A(7r)  and  S^,  is  to 


THE   ORTHORHOMBIC   POINT-GROUPS. 


13 


be  understood  as  a  two-fold  rotation  followed  by  a  reflection  in  the  hori- 
zontal plane.* 

Coordinates  of  equivalent  points,     xyz;        xyz;        xyz;        xyz. 

ORTHORHOMBIC  SYSTEM. 

The  orthorhombic  axes  of  reference  are  three  mutually  perpendicular 
axes  of  unequal  unit  lengths. 

Point-group  Cl- — Since  reflection  in  a  plane  containing  two  of  the  axes 
of  reference  and  normal  to  the  third  changes  the  sign  of  the  coordinate  value 
for  the  third  (confer  C2),  this  point-group  may  be  expressed  as 

Operations  of  sjmimetry:  1,     A(7r),        Sy,        SvA(7r). 

Sy  is  a  reflection  in  a  vertical  plane  (taken  through  Y  and  Z). 

Coordinates  of  equivalent  points:  xyz;        xyz;        xyz;        xyz. 


Fig.  12. 

Point-group  V. — The  rotations  about  the  three   mutually    perpendicular 
two-fold  axes  will  be  designated  as  U,  V,  and  W  (figure  12). 
Operations  of  sjTnmetry : 

1,       U,       V,       w. 

Coordinates  of  equivalent  points : 
xyz;        xyz;        xyz;        xyz. 

Point-group  V. — ^As  usual,  the  XY-plane  is  taken  as  the  horizontal  mir- 
roring plane. 

Operations  of  symmetry: 

1,       U,       V,       W,       S„       USb,       vs„       ws,. 
Coordinates  of  equivalent  points: 

xyz;        xyz;        xyz;        xyz;        xyz;        xyz;        xyz;        xyz. 

*  The  order  of  combining  the  operations  in  such  a  product  is  immaterial.     It  could  equally 
well  have  been  called  a  reflection  followed  by  a  two-fold  rotation. 


14 


THE   TETRAGONAL   POINT-GROUPS. 


TETRAGONAL  SYSTEM. 
The  three  tetragonal  axes  of  reference,  mutually  perpendicular  to  one 
another,  are  two  (the  X-  and  the  Y-axes)  of  equal  unit  length. 
Point-group  04  =  84. — 


Operations  of  symmetry: 

1,         S,a(^),         A(7r)*, 

Coordinates  of  equivalent  points : 


S.A  - 


(!) 


xyz;        yxz;        xyz;        yxz. 

Point-group  C4. — As  we  have  just  seen,  the  rotation  of  angle  -  about  the 

Z-axis  interchanges  the  X  and  Y  coordinates  and  leaves  the  new  X-coordi- 
nates  reversed  in  sign  (figure  13). 


-y 


yx 


\y 


Operations  of  symmetry: 
1,        A^|\        A(7r), 


Fig    13. 


AHf 


(W 


Coordinates  of  equivalent  points : 
xyz;        yxz;        xyz;        yxz. 

Point-group  V. — The  diagonal  reflecting  plane  contains  the  Z-axis  and 
bisects  the  angle  between  the  X-  and  Y-axis.    Reflection  in  such  a  plane 
(Sd)  interchanges  the  X-  and  Y-coordinates  (figure  14). 
Operations  of  symmetry. 

1,        U,        V,        W,         Sd,        USd,         VSd,        WSd. 

Coordinates  of  equivalent  points. : 

xyz;        xyz;        xyz;        xyz;        yx^;        yxz;        yxz;        yxz. 

*  This  arises  from  the  observation  that  two  reflections  in  the  same  plane  nullify  one  another. 


THE   TETRAGONAL   POINT-GROUPS. 


15 


Point-group  C4. — 

Operations  of  symmetry: 


1,     a(^^,     A(7r),     A^y),      S„     S^A(^y    S,A(x),    S,a(^^ 

This  may  be  more  conveniently  written  as  Cl=  {C4,  Sn},  signifying  that  the 
operations  of  C4  are  those  of  C4  plus  the  reflections  of  these  operations  in 
the  horizontal  plane.* 


Fig.  14. 

Coordinates  of  equivalent  points : 

xyz;  yxz;  xyz;  yxz;  xyz;  yxz;  xyz;  yxz. 
These  coordinates  illustrate  the  fact,  of  which  use  will  commonly  be  made 
in  the  work  which  follows,  that  a  two-fold  rotation  about  an  axis  combined 
with  a  reflection  in  a  plane  normal  to  the  axis  is  equivalent  to  an  inversion 
(figure  15).  Thus  C4={C4,  1}  is  an  alternative  expression  of  the  point- 
group  C4.  In  this  latter  case  the  coordinates  of  equivalent  points  would  be 
written  in  the  following  order. 


yxz; 


xyz; 


yxz; 


xyz;        yxz; 


xyz; 


yxz. 


xyz; 
Point-group  C4. — 

Operations  of  symmetry : 

C4  =    {  C4,     Sy  }  . 

Sv  is  again  a  mirroring  in  the  vertical,  YZ-plane. 
Coordinates  of  equivalent  points: 

xyz;        yxz;        xyz;        yxz;        xyz;        yxz;        xyz;        yxz. 
Point-group  D4. — The  four  two-fold  axes  lying  in  the  XY-plane  coincide 
with  the  X-  and  Y-axes  and  bisect  the  angles  between  them  (figure  6).     The 


*  In  the  future  this  abbreviated  representation  will  be  used  when  no  ambiguities  are  thereby 
introduced. 


16 


TETRAGONAL  AND   CUBIC   POINT-GROUPS. 


operations  of  symmetry  of  D4  may  consequently  be  obtained  by  appljdng 
the  operations  of  one  of  the  two-fold  axes  (the  one  coinciding  with  the  X- 
axis  will  be  employed)  to  those  of  C4. 
Operations  of  sjonmetry: 

D4={C4,U}. 

Coordinates  of  equivalent  points: 

xyz;        yxz;        xyz;        yxz;        xyz;        yxz;        xyz;        yx2. 
If  other  two-fold  axes  were  used,  the  order  of  the  last  four  coordinate  values 
would  be  changed. 


Fig.  15. 
Point-group  D4. — 

Operations  of  symmetry: 

DS={D4,S,}  =  {D4,I}. 
Coordinates  of  equivalent  points: 

xyz;        yxz;        xyz;        yxz;        xyz;        yxz;        xyz;        yKz; 
xyz;        yxz;        xyz;        yxz;        xyz;        yxz;        xyz;        yxz. 
CUBIC  SYSTEM. 
The  cubic  axes  of  reference  are  three  mutually  perpendicular  axes  with 
units  all  of  the  same  length. 


Point-group  T. — 
Operations  of  symmetry 

1, 


AHf 


U, 

Ai 


V, 
A21 


(t)'     A^f)'     ^{i} 


w, 

A.( 


CUBIC   POINT-GROUPS,  17 

A,  Ai,  A2,  As  are  rotations  about  the  trigonal  axes  of  a,  ai,  as,  at  of  figure  7. 
Coordinates  of  equivalent  points: 


xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx. 

Point  Group  T^.— 

Operations  of  symmetry : 

T»'={T,S,}  =  {T,I}. 

In  writing  the  coordinates  of  equivalent  points  the  second  of  the  representa- 
tions will  be  used. 

Coordinates  of  equivalent  points:    The  12  coordinate  positions  of  T,  and 

II. 

Point-group  T"*. — The  diagonal  mirroring  plane  is  taken  to  bisect  the  angle 
between  the  X-  and  Y-axes. 
Operations  of  symmetry: 

T«={T,S,}. 

Coordinates  of  equivalent  points:  The  12  coordinate  positions  of  T,  and 

III 

Point-group  O 


U, 


yxz;        yxz; 

yxz; 

yxz; 

xzy;        xzy; 

xzy; 

xzy; 

zyx;        zyx; 

zyx; 

zyx. 

of  symmetry: 

The  12  < 

)perati( 

/t> 

k          ^  / 

'37r\ 

U.,       B  (2^ 

)     M 

.^> 

/7r> 

L              / 

'Sir\ 

V„        6,(2^ 

)'     »■( 

j)' 

W2,      b/|^ 

),         B,( 

'3x\ 
.2/ 

Wx 

Rotations  about  the  various  axes  of  figure  8  are  represented  by  the  corre- 
sponding capital  letters.  The  four-fold  axes  b,  bi  and  bs  have  the  positions 
of  u,  V  and  w  of  figure  7. 

Coordinates  of  equivalent  positions:  The  12  coordinate  positions  of  T,  and 

IV. 


yxz; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

xzy; 

zyx; 

zyx; 

zyx; 

zyx.* 

*  The  order  of  writing  these  coordinates  has  been  changed  about  to  make  it  conform  with  its 
later  uses. 


18  CUBIC   AND   HEXAGONAL   POINT-GROUPS. 

Point-group  0^. — 

Operations  of  symmetry: 
0^={0,S,}  =  {0,I}. 
Coordinates  of  equivalent  positions:  The  48  coordinate   positions  of 
I,  II,  III,  and  IV. 

HEXAGONAL  SYSTEM. 
Rhombohedral  Division. 
The  point-groups  of  this  division  will  be  described  in  terms  of  two  kinds 
of  axes  of  reference.  The  rhombohedral  axes  (1),  all  of  the  same  unit  length 
and  making  equal  angles  with  one  another,  are  arranged  symmetrically 
about  the  three-fold  axis  (figure  16).  Two  of  the  hexagonal  set  of  axes  (2) 
are  of  equal  unit  lengths  and  make  an  angle  of  120°  with  one  another  (figure 
17) ;  the  third,  the  Z-axis,  is  of  a  different  unit  length  and  is  normal  to  the 
plane  of  the  X-and  Y-axes.  This  second  set  is  thus  a  special  case  of  the 
monoclinic  axes;  the  cubic  axes,  on  the  other  hand,  are  a  special  case  of  the 
rhombohedral  (1)  axes.  Coordinates  according  to  the  rhombohedral  axes  are 
given  below  under  I,  according  to  the  hexagonal  axes  under  II. 


Fig.  16. 

Point-group  C3. — 

Operations  of  symmetry: 

1,      a(|),      a(|). 

Coordinates  of  equivalent  points: 
I.     xyz;         zxy;  yzx. 

II.    xyz;         y-x,  x,  z;        y,  x-y,  z.* 

Point-group  C3. — 

Operations  of  symmetry: 

03={C3,I}. 


♦  A  reference  to  figure  17  will  show  the  source  of  these  coordinate  values. 


HEXAGONAL  POINT-GROUPS. 


19 


zxy; 
x-y,  X,  z; 


fzx. 
y,  y-x,  z. 


Coordinates  of  equivalent  positions : 

I.     xyz;  zxy;  yzx;  xy2; 

II.     xyz;      y-x,  X,  z;      y,x-y,  z;      xyz; 
Point-group  C3. 

Operations  of  symmetry:  Cl=  {C3,  Sy}. 
This  vertical  reflecting  plane  can  have  two  possible  positions,  one  containing 
both  the  X-  and  the  Z-axes  (hexagonal  axes  II),  the  other  containing  the 
Z-axis  and  a  line  in  the  XY-plane  which  makes  an  angle  of  30°  with  the  X-axis 
(see  figure  26).  The  reflection  in  a  plane  occupying  the  first  of  these  two 
positions  will  be  designated  as  Sa,  the  reflection  in  the  other  plane  by  Sg. 
Coordinates  of  equivalent  points : 


Fig.  17. 


xyz;       zxy; 


yzx; 


I. 

yxz; 

II. 


xzy; 


zyx. 


WhenCl={C3,  Sa}: 

xyz;        y-x,  x,  z;        y,  x-y,  z;        x-y,  y,  z;        yxz;        X,  y-x,  z. 

WhenCl={Ca,S3}: 

xyz;        y-x,  x,  z;        y,  x-y,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z. 
Point-group  D3. — 
Operations  of  symmetry:  D3=  {C3,  U}. 
The  two-fold  axis  of  rotation  may  lie  either  in  the  X-axis  or  in  a  line  in  the 
XY  plane  which  makes  an  angle  of  30°  with  the  X-axis.    A  rotation  about 
the  first-named  axis  will  be  called  Ua,  about  the  second,  Ug.     There  may  thus 
be  two  different  sets  of  coordinates  of  equivalent  points  for  the  point-group 
D3  corresponding  to  the  two  sets  already  defined  for  C3. 


20  HEXAGONAL   POINT-GROUPS. 

Coordinates  of  equivalent  points: 

I. 

xyz;        zxy;        yzx;        yxz;        xzy;        zyx. 

II. 

WhenD3={C3,  Ua}: 

xyz;        y-x,  X,  z;        y,  x-y,  z;        x-y,  y,  z;        yxz;        x,  y-x,  2. 

WhenD3-{C3,U«}: 

xyz;        y-x,  X,  z;        y,  x-y,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z. 

Point-group  D3. — It  can  be  shown  that  this  point-group  arises  from  the 
combination  of  D3  with  an  inversion.  Just  as  there  are  two  ways  of  ex- 
pressing D3  in  terms  of  hexagonal  axes  of  reference  (depending  upon  the 
position  of  the  two-fold  axis)  so  there  must  be  two  ways  of  expressing  Df. 

Operations  of  symmetry: 

Di={D3,Sd}  =  {D3,I}. 

Coordinates  of  equivalent  points : 

I. 


xyz; 

zxy; 

yzx; 

yxz; 

xzy; 

zyx; 

xyz; 

zxy; 

yzx; 

yxz; 

xzy; 

zyx. 

II. 

When  the  operation  of  the  two-fold  axis  is  Ua: 

xyz;        y-x,  x,  z;        y,  x-y,  z;        x-y,  y,  z;        yxz;        x,  y-x,  z. 
xyz;        x-y,  X,  z;        y,  y-x,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z. 

When  the  operation  of  the  two-fold  axis  is  Us : 

xyz;        y-x,  X,  z;        y,  x-y,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z; 
xyz;        x-y,  x,  z;        y,  y-x,  z;        x-y,  y,  z;        yxz;        x,  y-x,  z. 

Hexagonal  Division 
The  point-groups  of  this  division  of  the  hexagonal  system  will  be  expressed 
only  in  terms  of  the  hexagonal  axes. 

Point-group  C3. — 

Operations  of  symmetry: 

C§={C3,SU. 

Coordinates  of  equivalent  points : 

xyz;       y-x,  X,  z;       y,  x-y,  z;       xyz;       y-x,  x,  z;       y,  x-y,  z. 

Point-group  D3. — 
Operations  of  symmetry : 

D§={D3.  SJ. 


HEXAGONAL   POINT-GROUPS.  21 

Just  as  there  are  two  ways  of  expressing  D3,  so  there  will  be  two  ways  of 
stating  D3. 

Coordinates  of  equivalent  points: 
When  the  two-fold  axis  has  the  position  of  the  X-axis  (Ua) : 

xyz;        y-x,  x,  z;        y,  x-y,  z;        x-y,  y,  z;        yxz;        x,  y-x,  2; 
xyz;        y-x,  x,  z;        y,  x-y,  z;        x-y,  y,  z;        yxz;        x,  y-x,  z. 

When  the  two-fold  axis  makes  an  angle  of  30°  with  the  X-axis  (Us) : 

xyz;        y-x,  x,  z;        y,  x-y,  z;         y-x,  y,  z;        yxz;        x,  x-y,  z; 
xyz;        y-x,  x,  z;        y,  x-y,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z. 

Point-group  Ce- — 

The  operations  of  this  group  can  be  written  as  those  arising  from  the  opera- 
tion of  a  60°  axis  of  symmetry.     Taken  thus  the  operations  of  Cg  are: 
Operations  of  symmetry : 

1.      a(|),      a(|),      aw,      A(f),      a(|). 

Coordinates  of  equivalent  points: 

xyz;        y,  y-x,  z;        y-x,  x,  z;        xyz;        y,  x-y,  z;        x-y,  x,  z; 

Point-group  CI- — 

Operations  of  symmetry : 

C^={Ca,S,}. 

Coordinates  of  equivalent  points: 


xyz; 

y,  y-x,  z; 

y-x,  x,  z; 

xyz; 

y,  x-y,  z; 

x-y,  X,  z; 

xyz; 

y,  y-x,  z; 

y-x,  X,  z; 

xyz; 

y,  x-y,  z; 

x-y,  X,  z. 

Point-group  CI- — 
Operations  of  symmetry: 

Q={C6,Sv}.* 
Coordinates  of  equivalent  points: 

xyz;       y,  y-x,  z;        y-x,  x,  z;        xyz;       y,  x-y,  z;        x-y,  x,  z; 
X,  y-x,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z;        x-y,  y,  z;        yxz. 

Point-group  Df — 
Operations  of  symmetry: 

D«={Ce,U}. 

U  is  a  rotation  of  180°  about  axes  in  the  XY-plane,  one  of  which  coincides 
with  the  X-axis. 

Coordinates  of  equivalent  points: 

xyz;        y,  y-x,  z;        y-x,  x,  z;        xyz;        y,  x-y,  z;        x-y,  x,  z; 
X,  y-x,  z;        y-x,  y,  z;        yxz;        x,  x-y,  z;        x-y,  y,  z;        yxz. 

*  This  group  is  of  course  equally  the  result  of  operating  upon  Cg  by  a  two-fold  axis  coincident 
with  the  Z-axis.    That  is, 

C^={C3,Uil. 


22  HEXAGONAL  POINT-GROUPS — SPACE  LATTICES. 

Point-group  Dq. — 
Operations  of  symmetry: 

Dg={D6,S.}  =  {De,I}. 

Coordinates  of  equivalent  points. 

xyz;        y,  y-x,  z;        y-x,  x,  z;  xyz;        y,  x-y,  z;        x-y,  x,  z; 

X,  y-x,  z;        y-x,  y,  z;        yxz;  x,  x-y,  z;        x-y,  y,  z;        yxz. 

Xyz;        y,  x-y,  z;        x-y,  x,  z;  xyz;        y,  yx,  z;        y-x,  x,  z; 

X,  x-y,  z;        x-y,  y,  z;        yxz;  x,  y-x,  z;        y-x,  y,  z;        yxz. 

SPACE  LATTICES. 

A  series  of  parallel  planes  such  that  the  distance  between  any  two  consecu- 
tive planes  of  the  series  is  constant  is  called  a  set  of  planes. 

The  sum  total  of  the  points  of  intersection  of  any  three  sets  of  planes  is 
called  a  regular  space  lattice. 

z 


Fia.  18.  A  sjrmmetrical  lattice.    The  intersection  points  of  this  figure  are 
points  of  the  lattice. 

If  some  point  of  a  lattice  (0  of  figure  18)  is  taken  as  the  origin  of  coordi- 
nates, the  neighboring  points  of  the  lattice  are  given  by  the  translations 
±2t„  ±2Ty,  ±2r,  along  the  X,  Y  and  Z  axes;  and  in  general  any  point 
of  che  lattice  is  given  by  the  composite  translation 

Ti  =  d:  2mT  j  =t  2nTy  ±  2pT, 

where  m,  n  and  p  are  any  integers  or  zero.    The  three  translations,  2tx,  2Ty, 
2t„  giving  neighboring  points  of  the  lattice,  are  called  the  primitive  trans- 


THE    14   SPACE   LATTICES. 


23 


lations.  It  is  customary  to  define  a  lattice  by  stating  its  primitive  trans- 
lations with  respect  to  the  axes  of  reference.*  This  definition  is  sufficient 
since  the  primitive  translations  of  a  lattice  can  be  considered  as  those  transla- 
tions which  will  yield  all  of  the  points  of  the  lattice  by  their  continued 
appUcation,  first  to  a  point  of  the  lattice  chosen  as  origin,  and  to  the  new 
points  continually  derived  from  this  and  succeeding  applications. 

It  can  be  shown  that  but  fourteen  symmetrical  lattices  are  possible;  each 
of  them  has  the  complete  symmetry  of  one  of  the  seven  systems  of  crystallo- 
graphic  symmetry  (counting  the  rhombohedral  division  of  the  hexagonal 
system  as  a  separate  system). 

The  primitive  translations  of  these  14  space-lattices,  identical  with  the 
lattices  of  Bravais,  are  as  follows.  The  axes  of  reference  are  the  same  as 
those  used  for  the  point-groups  of  corresponding  symmetry. 

Primitive  translations. 


2r,;2ry;2T.. 
2r,;2Ty;2r,. 

2r,;  2ry;  2t,. 

'"x)  ■'■yJ  Tx>  "TyJ  2t,. 
•^"^X)  ''yj  ^z'>  "^7)  ''z* 
''yj  "^ty  ''«'  ''x>  ''x>  '''y 

2Tx;    2Ty]    2r^',    Tjj,    Ty,    T,. 

2r,;2Ty;2r,. 

Tij  Ty'y  ^x)  "'''yl  2t,. 
"^yy  "^zj  "^zy  '^xy  "^xy  "^y 

2tx;  2Ty;  27^;  t,,  Ty,  r,. 


Symbol. 

Tridinic  system. 

1. 

Ttr 

Monoclinic  system. 

2. 

r„. 

3. 

r.' 

Orthorhombic  system. 

4. 

To 

5a. 

To'  (a) 

b. 

To'  (b) 

6. 

To" 

7. 

To'" 

Tetragonal  system. 

8a. 

Ft  (a) 

b. 

rt(b) 

9a. 

r/  (a) 

b. 

Ft'  (b) 

Cubic  system. 

10. 

r. 

11. 

Tc' 

12. 

Tc" 

Hexagonal  system. 

13. 

Trb 

14. 

Tu 

2T,;2Ty;2T,. 

'^yy  '^z'y  ^2>  '^x'y'^xi  "^y 

2t^;  2Ty;  2t,;  t,,  Ty,  t,. 


2tx  ;  2Ty ;  2tj .     (Rhombohedral  Axes) 
2tx;  2Ty;  2t^.     (Hexagonal  Axes) 


♦  Different  groups  of  primitive  translations  for  a  single  lattice  are  possible  by  taking  the  unit 
directions  differently.  We  shall  have  use  for  the  primitive  translations  just  defined  and  for  no 
others. 

t  By  Ty,  Tz  is  meant  a  translation  Ty  along  the  Y-axis  followed  by  one  of  length  Tz  along  the 
Z-axis.  The  translation  Ty,  —  Tz  is  similar  except  that  Tz  is  here  taken  in  the  —  z  direction. 
These  are  written  by  Schoenflies  as  Ty  +  Tz  and  Ty  —  tz  respectively. 


24 


THE  NATURE   OF  SPACE-GROUPS. 


Lattices  13  and  14  belong  to  the  rhombohedral  division;  lattice  14  has  the 
complete  symmetry  of  the  hexagonal  division  of  the  hexagonal  system. 

Lattices  2,  4,  8a,  10,  13  and  14  are  all  special  cases  of  lattice  1,  in  which  the 
lengths  of  the  units  along  the  axes  or  the  angles  between  the  axes  have  par- 
ticular values.  The  lattices  having  the  symmetry  of  the  tetragonal  and  of 
^he  cubic  system  can  be  looked  upon  as  special  cases  of  the  orthorhombic 
space  lattices;  in  this  process  of  speciaUzation,  for  lattices  of  tetragonal  sym- 
metry, if  the  axes  are  taken  after  the  manner  of  lattice  4,  8a  is  obtained,  if 
according  to  5,  8b  results.  The  two  forms  of  8  are,  however,  identical. 
In  a  similar  fashion  9a  and  9b  arise  from  6  and  7. 

SPACE-GROUPS. 

In  giving  analytical  representations  to  each  of  the  32  point-groups  the 
different  ways  have  been  expressed  in  which  points  can  group  themselves 
about  a  central  position  so  that  the  aggregate  of  points  will  by  their  arrange- 
ment exhibit  crystallographic  symmetry.    We  are  not,  however,  primarily 


Fig.  19.   The  point-group  Cl-    The  points  P,  P/,  P//,  P///  are  the  four 
equivalent  points  of  this  point-group. 


interested  in  such  an  aggregate  of  points  about  a  single  position  in  space  but 
rather  in  the  indefinite  extension  in  all  directions  of  such  a  symmetrical 
grouping  of  points.  In  order  to  accomplish  this,  it  is  necessary  to  distribute 
point-groups  (or  perhaps  other  suitably  symmetrical  groupings  of  points), 
properly  oriented  according  to  some  regular  pattern  which  repeats  itself 
indefinitely  in  all  directions.  Such  a  regular  pattern  must  be  one  of  the  14 
space  lattices.  The  indefinitely  extended  symmetrical  arrangement  of  points 
all  equivalent  to  one  another,  which  is  obtained  by  placing  such  groups  of 


A  TYPICAL  SPACE-GROUP,   Ca. 


25 


equivalent  points  with  their  centers  at  the  points  of  one  of  the  regular  space 
lattices,  is  a  space-group.* 

For  the  sake  of  illustration  the  very  simple  space-group  which  is  obtained 
by  placing  the  point-group  Cg,  the  holohedry  of  the  monoclinic  system,  (figure 
19)  at  the  points  of  the  monocUnic  space  lattice  Tm  will  be  considered,  f  A 
portion  from  this  space-group  is  shown  in  figure  20.     The  four  equivalent 


Fig.  20.  A  portion  from  the  monoclinic  space-group  C^u- 


points  P,  P/,  P//  and  P///  (and  the  two-fold  axis  of  symmetry  and  the  plane 

normal  to  it)  of  C2  repeat  themselves  about  each  of  the  points  O,  A,  B 

of  the  first  monoclinic  lattice  Fm.  Taking  O  as  the  origin,  then  the  coordin- 
ates of  the  points  of  the  group  about  A,  the  point  of  the  lattice  obtained  by  the 
primitive  translation  2tx,  are 


x-f-2Tx,  y,  z;  2t^-x,  y,  z;  2r^-x,  y,  z;  x-F2t„  y,  z. 

In  a  similar  way  the  coordinates  of  the  equivalent  points  about  the  other 
neighboring  points  of  the  lattice  and  in  general  about  any  point  of  the  lattice 

*  The  view  which  one  takes  of  a  space-group  will  depend  largely  upon  his  interests.  For 
instance,  the  crystallographer  will  in  all  probability  consider  a  point-group  as  a  particular  aggre- 
gation of  elements  of  symmetry  arranged  in  some  definite  fashion.  The  space-groups  will  then, 
first  and  above  all,  describe  to  him  the  way  in  which  these  elements  of  symmetry  can  be  dis- 
tributed throughout  a  crystal.  On  the  other  hand,  the  physicist  or  chemist  who  is  accustomed 
to  think  of  a  crystal  essentially  as  an  orderly  arrangement  of  atoms  or  molecular  groupings  of 
atoms  will  probably  incline  to  the  more  analytical  view  of  point-groups  and  space-groups  as  aggre- 
gates of  equivalent  points  which  are  potential  positions  for  the  atoms  in  crystals.  Because  we 
are  interested  here  in  discussing  only  those  phases  of  the  theory  of  space-gioups  which  are  of 
immediate  use  to  the  physical  study  of  the  structures  of  crystals,  the  characteristics  of  symmetry 
possessed  by  the  various  space-groups  will  receive  only  such  treatment  as  is  required  for  the 
building  up  of  an  analytical  expression  of  the  results  of  the  theory. 

t  Figure  18  will  illustrate  Fm  if  X  and  Y  have  any  unit  lengths  and  make  any  angle  with  one 
another,  and  if  Z  is  norma!  to  the  plane  XY. 


26  THE   TRICLINIC   SPACE-GROUPS. 

are  given  by  one  of  the  following  sets  which,  taken  together,  completely 
define  this  space-group: 

x±2mT,,  y±2nTy,  z±2pr,; 

±2mTx— X,  ±2nTy— y.  z±2pT,; 

±2mTx  — X,  ±2nTy— y,         ±2pT,  — z; 

x±2mTx,  y±2nTy,               ±2prj  — z; 

where,  as  before,  m,  n  and  p  can  be  any  integers  or  zero. 

Some  of  the  space-groups  are  obtained  by  thus  placing  point-groups  at  the 
points  of  the  lattice  of  corresponding  symmetry;  the  rest  of  the  230  typical 
ways  of  arranging  points  so  that  the  assemblage  will  exhibit  crystallographic 
symmetry  may  be  obtained  by  placing,  at  the  points  of  these  lattices,  groups 
of  points  analogous  to  the  point-groups,  and  derived  from  them,  of  such  a 
nature  that  the  symmetry  of  the  aggregate  is  that  of  one  of  the  point-groups 
themselves. 

It  is  obvious  that  a  space-group  is  completely  defined  (analytically)  when  the 
coordinates  of  the  equivalent  points  ranged  about  one  point  of  the  lattice 
(the  points  of  a  point-group  or  of  a  "modified  point-group")  and  the  primitive 
translations  of  the  lattice  are  given;  for,  as  we  have  just  seen  in  the  case  of 
the  monochnic  space-gi"Oup,  with  this  information  it  is  always  possible  to 
reconstruct  the  space-group. 

AN  OUTLINE  OF  THE  DERIVATION  OF  THE  SPACE-GROUPS. 

The  nature  of  each  of  the  space-groups  will  be  apparent  from  the  following 
tabular  outline.  Under  each  class  of  symmetry  a  brief  discussion  of  the 
development  of  the  space-groups  exhibiting  its  symmetry  will  be  given.  This 
will  be  followed  by  a  statement  under  three  headings  of  (1)  the  symbol  of 
the  space-group,  (2)  an  abbreviated  indication  of  its  particular  derivation,  and 
(3)  the  fundamental  lattice  underlying  it. 

TRICLINIC  SYSTEM. 
Hemihedry. — 

The  single  space-group  of  this  class  is  obtained  by  placing  the  single  equiva- 
lent point  of  the  point-group  Ci  at  the  points  of  the  lattice  Ftf. 

1.  {C}=a,r,r}.*  T^ 

Holohedry. — 

The  single  space-group  having  this  symmetry  is  obrained  by  placing  the 
equivalent  points  of  Q  at  the  points  of  the  lattice  Ta. 

2.  c|  =  {c.,r,,}.  r„ 

*  The  space-group  symbol  is  a  simple  adaptation  of  the  symbols  used  for  the  point-groupa. 
The  letters  to  be  found  in  exponent  position  in  the  symbols  for  point-groups  are  reduced  to  the 
subscript  position.  The  different  space-groups  isomorphous  with  a  particular  point-group  are 
distinguished  by  numbers  in  the  exponent  position.  Thus  Cj^j^  is  the  hfth  space-group  (isomor- 
phoua  with  the  point-group  C^)  that  is  defined. 


3. 

cl  = 

4. 

c^= 

5. 

c^= 

6. 

ct= 

THE   MONOCLINIC   SPACE-GROUPS.  27 

MONOCLIJIC  SYSTEM. 
Hemihedry. — 

The  space-groups  having  this  symmetry  can  be  developed  by  combining 
the  space-group  C|  when  it  has  the  speciahzed  form  of  either  Tm  or  Fm, 
with  a  ghding  reflection  in  a  plane  which  is  taken  as  that  of  the  X-  and  Y- 
axes.* 

|r„„S,(r)).  F„ 

1  Fna  »  ^h  )  •  Fm 

Hemimorphic  hemihedry. — 

Since  the  point-group  C2  is  obtained  by  combining  Ci  with  a  two-fold 
axis,  the  space-groups  isomorphous  with  C2  can  be  obtained  by  combining 
the  lattices  Fn,  and  F^'  with  screw  axes  of  symmetry.  The  translation  com- 
ponents of  these  screw-axes  are  either  zero  or  half  a  primitive  translation  in 
the  direction  of  the  Z-axis. 

7.  C^={F„,A(7r)}.  r^ 

8.  Ci  =  {F„,  A(7r,  T.)}.  F„, 

9.  C^  =  { F„',  ACtt)  I  =  { F„.',  A(7r,  Tr) } .  F„.' 

Holohedry. — 

The  space-groups  isomorphous  with  Cj  can  be  obtained  by  multiplying 
(= combining)  space-groups  isomorphous  with  C2  with  the  operation  of  a 
glide  plane  of  symmetry.  Since  a  rotation  of  180°  combined  with  a  reflec- 
tion in  a  plane  at  right  angles  to  the  axis  of  rotation  is  equivalent  to  an 
inversion,  these  space-groups  result  also  from  multiplying  the  groups  iso- 
morphous with  C2  by  an  inversion. 


10. 

C2h=  {C2,  Sh}. 

F, 

11. 

C2h  —  1  C2,  feh  } . 

F, 

12. 

C2h=  (C2,  Shj. 

F, 

13. 

C2l,=  {CLS,(r)}. 

F, 

14. 

ci^{ci,s^(t)}. 

F, 

15. 

CA={Cf,S,(r)}. 

F, 

ORTHORHOMBIC  SYSTEM. 

Hemimorphic  hemihedry. — 

The  intersections  with  the  XY-plane  of  the  axes  of  space-groups  C™  (the 
space-groups  having  the  symmetry  of  C2)  when  the  angle  between  the  axes 
has  the  special  value  of  90°,  is  given  by  the  points  A,  B,  C,  D,  Ai  .  .  .  . 
of  figure  21.    The  space-groups  isomorphous  with  Cg  can  be  developed  j^y 

*  A  glide  plane  the  translation  component  of  which  is  zero  is  of  course  a  simple  reflecting 
plane,     r,  a  primitive  translation  in  the  XY-plane,  may  then  be  chosen  as  either  t^  or  xy. 


28 


ORTHORHOMBIC   SPACE-GROUPS. 


multiplying  groups  ismorophous  with  Ca  by  a  vertical  glide  plane  of  sym- 
metry, that  is,  one  parallel  to  or  containing  the  Z-axis.  The  various  possible 
positions  of  the  intersections  of  these  planes  with  the  XY-plane  are  shown 
by  c,  o-m,  etc.  of  figure  21a  and  c^,  a^',  etc.  of  figure  21b. 


zzy 


A    d 

B 

A, 

6  m. 

6  nix 

^1 
C 

d' 

di 

D 

^ 

^3 

1 

16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 
34. 
35. 
36. 
37. 


ZTX 


Fig.  21. 


r*  - 

p2   _ 

^2v  — 

C4 
2v  = 
5 


^2v  — 


|CLS}  =  {CLSi(rJi 

|CLS(r,)} 

1C2,  S(tJ}  =  {C2,  Sm 

[CLS(rJ}. 


^2v 

*^2v 

p9  _ 
V-'2v  — 


=  {C2,  S(Tx  +  tJ}. 

=  {CiS(r,+r,)}. 

=  {C2,  Sni(Tx)}. 


1C2,  Sni(Tx)}. 
{C2,  Sni(Tx+Tz)}. 
{C2,  Sd}  =  {C2,  Sflij 
{C2,  Sd}. 


C2v=  {C2,  Sdfrj,)}. 
C'2v  ~  {  ^2>  ^ )  • 
C2v={C2,  S(tJ}. 

S(rJ}. 


C^'  =  {C 


V^2V 
^2v' 


{C2^s}. 

{C2,  Sin[§(l 


=+rJ]}. 


''2> 


^2V 

^22 

-'2V 


3    ^•^^' 

2>  SdCrJ}. 


=  {C2,  Sd(Tj}. 


KO' 


To 

To 

To 

To 

To 

To 

To 

To 

To 

To 

ro'(a) 

ro'(a) 

ro'(a) 

r«'(b) 
ro'(b) 
ro'(b) 
ro'(b) 

To" 

rin 
« 

rin 
0 

rtn 
0 


erA: 


Enantiomorphic  hemihedry. — 

Definition. — If  a  certain  portion  of  the  operations  of  a  group  when  taken 
alone  themselves  form  a  group,  they  define  a  sub-group. 

The  space-groups  isomorphous  with  the  point  group  V  are  best  described 
by  giving  the  sub-groups  whose  axes  are  parallel  to  the  X,  Y-  and  Z-  axes 
of  the  lattice  (and  of  the  coordinates). 


ORTHORHOMBIC   SPACE-GROUPS. 


29 


38. 

Y'={Cl,Cl,Cl}.    To 

39. 

Y'={Cl,Cl,Cl}.    To 

40. 

Y'^{Cl,Ci,Cl\.    To 

41. 

Y'=={Cl,ClCl\.     To 

42. 

Y^={Cl,ClCl}.     ro'(a) 

43. 

V«={CLCLC^}.     ro'(a) 

44. 

Y'=={ClCl,Cl\.     To" 

45. 

V«=lC^,C,^q}.     To'" 

46. 

Y'={ClCl,Cl\.     To'"* 

Holohedry. 

— 

The  space-groups  isomorphous  with  V^  can  bo  obtained  by   combining 

groups  isomorphous  with  V  with  a  horizontal  ghding  reflection.     It  is  more 

simple,  however,  to  consider  them  as  developed  by  combining  certain  groups 

V™  with  inversions.     The  locations  of  these  points  of  inversion  will  be  clear 

from  a  reference  to  figure  22. 

47. 

Vi={VMl.       To 

48. 

V^={VM^}.     To 

49. 

V^={VM.}.     To 

50. 

V^={VM,1.     To 

51. 

V^={V^I}.       To 

52. 

V«={VM„}.     To 

53. 

V^={VMJ.     To 

54. 

V^={VM,}.     To 

55. 

V«={VM}.       To 

56. 

V^.°={VM^}.    To 

57. 

Vy  =  {V',I,}.     To 

58. 

V^h'={VMw}.     To 

59. 

V\f={VM,}.     To 

60. 

V\,^={VM,}.      To 

61. 

Y'^^{Y\\\.       To 

62. 

V^,«={VM,}.     To 

63. 

V»J={VM}.      ro'(a) 

64. 

V^.'={VMJ.     ro'(a) 

65. 

V^,«={V«,I}.      ro'(a) 

66. 

Vl''={V«,I„,}.     ro'(a) 

67. 

VV={V«,  IJ.     ro'(a) 

68. 

Vi^  =  {V«,I,}.     ro'(a) 

69. 

Vi^^lV,  I}.       To" 

70. 

C1*={V,I^}.     To" 

71. 

V=if={V«,I}.      To'" 

72. 

Vl«={VM„}.     To'" 

73. 

VY={VSI}.       To'" 

74. 

Vl«={V«,I,}.     To'" 

*  These  two  last  space-groups  differ  in  the  manner  of  distribution  of  their  axes.     For  the 
former  the  axis  of  rotation  lies  in  the  line  AD,  for  the  latter  in  the  line  BC  of  Figure  21. 


30 


TETRAGONAL   SPACE-GROUPS. 


TETRAGONAL  SYSTEM. 

Tetartohedry  of  the  second  sort. — 

The  groups  Sf  can  be  obgained  by  combining  groups  isomorphous  with  Cj 
with  a  rotary-reflection  (a  rotation  combined  with  a  reflection)  having  the 
same  axis  as  the  group  C^. 

75.  s1={CLa}.    r, 

76.  SI={C|,A}.    r/ 
Hemihedry  of  the  second  sort. — 

The  space-groups  isomorphous  with  V*  can  be  obtained  by  multiplying 
groups  isomorphous  with  V  by  the  operation  of  a  diagonal  vertical  glide  plane 
of  symmetry.  A  reflection  in  the  plane  WMGA  of  figure  22  will  be  called 
(Td,  one  in  the  parallel  plane  through  F,  0-^,. 


^   , 

\                                                            y/ 

/ 

Ay 

j>Tr 

y 

X-H-      -H 

^ 

-y 

1 

/ 

rx- 

/ 

/ 

U ^ir 

/                       yo 

'--7% 

/                  y 

y  / 

/             y 

yy  . 

r 

Fig.  22. 

77.     V^={VSS,}.                                     Te 

78.     V,^={VSS,(rJ}                                r» 

79.     Va'={V^S,}.                                 r» 

80.     V^={V3,S,(rJ}.                           r. 

81.  v^={V6,s,}.                      r* 

82.    V«={VSS,(r.)}.                          r. 

83.   v^=|v«,s..(^^)}           r, 

84.     V«=|ve,S^(^^^+r.)|        T, 

85.    V,»={V7,S,}.                             r/ 

86.    ¥^^={^,8,(7,)}.                        v: 

87.   vy  =  {V8,s,)}.                      r/ 

88.    Vl,='={V»,S,(Tr)}.*                      r/ 

\2'  2'  2j'    '  ~\      2'2'2/''~V2'       2'2j''     ~V2'2'        2/ 


TETRAGONAL   SPACE-GROUPS. 


31 


Tetartohedry. — 

The  space-groups  C™  can  be  derived  by  arranging  screw-axes  of  s)nmmetry 
according  to  the  two  tetragonal  lattices. 


89.     Cl  = 


90. 


91. 


92. 


93. 


94. 


i={A(0r.} 
cl={A(-;,|'),r.}. 

Cl={A(||-),r.}. 

cf={AQ,r.'}. 

cS={A(|f),r.'} 


Paramorphic  hemihedry. — 

The  groups  C^  are  most  readily  obtained  by  inverting  groups  isomorphous 
with  C4  either  through  a  point  lying  in  a  four-fold  axis  or  midway  of  a  line 
joining  two  four-fold  axes.     This  second  inversion  will  be  represented  by  Ii. 

Tt 
.      Ft 

.    r, 

Ft' 

Hemimorphic  hemihedry. — 

The  groups  C^  are  obtained  by  multiplying  groups  C"  by  vertical  gliding 
reflections.     The  positions  of  these  reflecting  planes  are  shown  in  figure  23. 


95. 

CA  =  {C1,I}. 

96. 

^4li  —  1^4)  !}• 

97. 

CA  =  {Cl,Ix} 

98. 

CA={CU,} 

99. 

C4h=  {C4,  I}. 

100. 

C4h=  {C4,  II} 

101. 

C4V=    {C4,     Sg}. 

Ft 

102. 

C4v=  {C4,  So}. 

Ft 

103. 

C4v=  {C4,  Sg}. 

Ft 

104. 

C/v={Ct.Se}. 

Ft 

105. 

C/v={Cl,S,(rJ}. 

Ft 

106. 

C4^={CLSe(r.)}. 

Ft 

107. 

C4;={Ct,S3(rJ}. 

Ft 

108. 

C4v=  {C4,  Se(Tj)}. 

Ft 

109. 

C4v=  {C4,   Sg}. 

Tt' 

110. 

Clv°={Cps(rJ}. 

Tt' 

111. 

C4v=  {C4,  Sc}. 

Tt' 

112. 

Cl^={aSe(rJ}. 

Tt' 

32 


TETRAGONAL   SPACE-GROUPS. 


Enanliomorphic  hemihedry. — 

Since  the  point-group  D4  results  from  the  multiplication  of  C4  by  a  two-fold 
axis  lying  in  the  plane  normal  to  the  four-fold  axis  of  C4,  the  groups  D^n 
arise  by  multiplying  certain  of  the  groups  C tn  by  two-fold  axes  lying  in  the 
XY-plane.  The  positions  of  these  axes  are  shown  in  figure  23,  if  the  Hnes 
AB  and  C1C2  define  the  axes  Ug  and  Uc  respectively. 


Fig.  23. 


113. 

Dl={Cl,UJ. 

Ft 

114. 

Dl={Cl,Ue}. 

Ft 

115. 

T>l={Cl,V,\. 

Ft 

116. 

T>t  ={ClVJ. 

Ft 

117. 

D|={CtUJ. 

Ft 

118. 

D^={Cf,Ue|. 

Ft 

119. 

Dl={Ct,UJ. 

Ft 

120. 

D|={CtUe}. 

Ft 

121. 

DS={C|,  UJ. 

Tt' 

122. 

D^4°={C«,  U.}. 

Tt' 

Holohedry- 

- 

The  space-groups  D^  may  be  deiived  by  combining  groups  of  D^  with  an 
inversion.  If  the  axes  striking  the  XY-plane  in  A,  Ai,  etc.  (figure  23)  are 
called  o  and  those  meeting  the  plane  in  points  corresponding  to  B  are  called  b. 
then  the  points  of  inversion  are  located  (1)  at  the  intersection  of  a  with  an 
axis  parallel  to  U„  (2)  midway  between  two  such  points  of  intersection,  (3)  on 
an  axis  parallel  to  Ub,  midway  between  a  and  b  or  (4)  half  of  the  way  between 
a  and  b  and  half  way  between  axes  parallel  to  Ug.  The  inversions  through 
these  four  points  will  be  denoted  by  I,  I',  Ii  and  Ii'.    These  four  inversions 


TETRAGONAL  AND    CUBIC   SPACE-GROUPS. 


33 


are  equivalent  to  inversions  I,  I, 
G,  and  M  of  figure  22. 


Igj  and  Ija  about  the  four  points  A,  W, 


123. 

DA  = 

Dl,Il 

Ft 

124. 

DA  = 

Dl,I. 

!.      Tt 

125. 

DA  = 

DL  IJ 

Tt 

126. 

DA  = 

Dl,I„, 

}•      Tt 

127. 

Dil}. 

Ft 

128. 

DA  = 

DLlw 

1-       Tt 

129. 

d;,= 

Bl,  I J 

Tt 

130. 

^4h  — 

DMn, 

|.      Tt 

131. 

n^  — 

Dtl} 

Ft 

132. 

-nio_ 

J^4h  — 

Dtlw 

!•       Tt 

133. 

n^^- 

^4b  — 

D|,IJ 

•       Tt 

134. 

J-'4h  — 

D|,  In, 

}•      Tt 

135. 

^4h  — 

Dl,I}. 

Ft 

136. 

^4h  — 

Dt  I. 

[.      Tt 

137. 

^4h  — 

D^  I,} 

.       Tt 

138. 

^4h  — 

DMn. 

}•      Tt 

139. 

n^^  — 

D4M}. 

Tt' 

140. 

r)i8_ 

J-'4h  — 

D^  Iw 

!•     r/ 

141. 

-ni9_ 

^4h  — 

D^",  I« 

I.    r/ 

142. 

^4h  — 

D^.°.  I„ 

J.    r/ 

CUBIC  SYSTEM. 

Tetartohedry. — 

The  space-groups  isomorphous  with  T  can  be  obtained  by  combining  certain 
groups  V™  with  the  operation  of  a  three-fold  rotation  axis.  Except  in  the 
case  of  the  group  derived  from  V,  when  it  must  be  AA',  the  position  of  this 
three-fold  axis  can  be  that  of  any  diagonal  of  figure  22.  This  rotation  of 
angle  /  will  be  represented  by  A. 


143. 
144. 
145. 
146. 
147. 


T»={VS  A}. 
T2={V^,  A}. 
T3={V8,  A}. 
T4={V^  A}. 
Ts={V«,  A}. 


Paramorphic  hemihedry. — 

Since  the  point-group  Tjj  can  be  derived  from  the  point-group  T  by  com- 
bining it  with  an  inversion  (as  well  as  with  the  operation  of  a  horizontal  plane 
of  symmetry),  the  groups  isomorphous  with  T^  can  be  obtained  from  the 
groups  T™  by  combining  them  with  an  inversion.     This  center  of  symmetry 


34 


CUBIC   SPACE-GROUPS. 


lies  either  at  a  corner  of  the  cube  of  figure  22  (A)  or  at  M. 
sions  will  be  called  I  and  1^  respectively. 


These  two  inver- 


148. 
149. 
150. 
151. 
152. 
153. 
154. 


Tl  = 
T^  = 
T^  = 
T^  = 

n= 

^7—  ( 


TM}. 

T2,I}. 

T3,I}. 
TSI}. 

T^={T^I}. 


Hemimorphic  hemihedry. — 

The  groups  isomorphous  with  T^  can  be  derived  by  combining  groups  T™ 
with  a  gliding  reflection  in  a  diagonal  plane.  This  plane  can  be  taken  as 
WMGA  of  figure  22. 


/ 

/ 

/ 

A. 

/ 

> 

/ 

Hi 

Y 

/ 

/ 

z. 

) 

y- 

/ 

4 

} 

/ 

/ 

1/ 

/ 

Fig.  24. 


Fig.  25. 


155. 

Tl={TSS,}. 

To 

156. 

T^={T2,Sd}. 

To' 

157. 

T^={T3,S,}. 

To" 

158. 

T^={TSS,(r)}. 

To 

159. 

T,^={TSS,(r)}. 

r  ' 

*■  c 

160. 

T^={T^S,(r)|. 

r  " 

^  c 

Enantiomorphic  hemihedry. — 

The  groups  0™  result  from  combining  groups  T™  with  the  operation  of  a 
two-fold  rotation  axis.  This  axis  may  be  taken  parallel  to  UK  of  figure  22. 
If  it  passes  through  the  point  M  of  figure  22  the  rotation  will  be  denoted  by 
Urn,  (2)  if  it  has  a  parallel  position  through  the  point  A  by  U,  (3)  if  it  hes  in  the 
line  bisecting  AM  (see  figure  24)  by  Ui,  or  (4)  if  it  bisects  MA'  by  Us. 


CUBIC   AND   HEXAGONAL   SPACE-GROUPS.  35 

161.  0^={TSU}.  Te 

162.  02={TSU„}.  Te 

163.  03={'P,  U}.  r/ 

164.  0*={T2,  U„}.  Te' 

165.  0«={T3,  U}.  Tc" 

166.  06={T*,  Ui}.  Tc 

167.  0'={T*,  U2I.  Te 

168.  08={T5,  U}.  To" 

Holohedry. — 

Since  the  point-group  O^  results  from  O  by  the  operation  of  a  center  of 
symmetry,  as  well  as  of  a  horizontal  reflecting  plane,  the  groups  O"  iso- 
raorphous  with  O'*  can  be  obtained  by  combining  groups  0™  with  an  inversion. 
These  centers  may  be  at  A,  A',  M  or  M'  of  figure  25;  the  corresponding  inver- 
sions will  be  called  I,  I',  Im,  Im'. 


169. 

Oi  =  {OSI}. 

Tc 

170. 

Og={OSI„,}. 

Tc 

171. 

0^={0^I| 

To 

172. 

0^  =  {0M^}. 

r„ 

173. 

0^={0'  I}. 

To' 

174. 

0«={03,I'}. 

To' 

175. 

0^  =  {0M^}. 

To' 

176. 

Og={OM:„}. 

To' 

177. 

0«  =  {0M}. 

To" 

178. 

0^t"-{0«,I}. 

To" 

HEXAGONAL  SYSTEM. 
RHOMBOHEDRAL  DIVISION. 


Tetartohedry. — 


The  space-groups  isomorphous  with  C3  can  be  obtained  by  combining 
the  lattices  Th  and  Trh  with  a  three-fold  screw  axis.  The  translation  com- 
ponent of  this  screw-motion  is  to  be  taken  along  the  Z-axis. 


179.  c^=|A(|'),r,|.  r„ 

180.  C|  =  |A^y,y),rb|.  r^. 

181.  c^  =  {A(|,|'),r,}.  r, 

182.  Ct  =  ^A(~yT,^  Trt 


36 


HEXAGONAL   SPACE-GROUPS. 


Paramorphic  hemihedry. — 

The  two  space-groups  Cs™  can  be  obtained  by  combining  groups  C™  with 
an  inversion  (I). 

183.  C3\  =  {CLI}.    r^ 

184.  Cl  =  {Ct,l}.     Tr, 

Hemimorphic  hemihedry. — 

The  vertical  reflecting  plane  will  contain  the  vertical  (Z)  axis  and  either 
(1)  the  X-axis — of  the  point  and  isomorphous  space-group — (A A'  of  figure 
26),  or  (2)  a  line  (AB  of  figure  26)  which  Ues  in  the  XY-plane  and  makes  an 
angle  of  60"  with  the  X-axis.  In  the  first  case  the  reflection  will  be  desig- 
nated So,  in  the  second  S.. 


^x 


Fig.  26. 
185.     C3V={  0^,83}.  T^ 

186.  C3l  =  {cLsj.       r, 

187.  C3l  =  {cLS3(r,)}.   r, 

188.     C3t={CLS.(r,)}.     r, 

189.  C3^  =  {c*.Sa|.       r,^ 

190.  C3^  =  {c^.s,(tJ}.   r,, 

Enantiomorphic  hemihedry. — 

The  space-groups  D"  result  from  operating  upon  groups  C™  with  a  two- 
fold axis  which  has  the  position  either  of  AA'  of  figure  26,  (11^),  or  of  AB, 

(Ue). 


191. 

D^={CLUJ. 

192. 

DI  =  {C^,UJ. 

193. 

T>l  =  {Cl,VJ. 

194. 

D^={Ci,  UJ. 

195. 

D|={C3^UJ. 

196. 

D!={C3^UJ. 

197. 

D^={C3^U3}. 

HEXAGONAL   SPACE-GROUPS. 


37 


Holohedry. — 

The  groups  D^  are  most  easily  obtained  by  combining  groups  of  D™  with 
an  inversion.  This  point  of  inversion  will  lie  either  at  the  intersection  of  a 
three-fold  and  a  two-fold  axis,  (I),  or  midway  between  two  such  intersec- 
tions (I'). 


198. 
199. 
200. 
201. 
202. 
203. 


D3^,  =  {DJ,I}. 


D/,  =  {D 


LI) 


D3t.  =  {DLr) 


Trh 


HEXAGONAL   DIVISION. 

Trigonal  paramorphic  hemihedry. — 

The  single  space-group  isomorphous  with  C3  is  obtained  by  reflecting  Cj 
in  a  horizontal  plane. 

204.    C/.=  {CLS,}.    r. 

Trigonal  holohedry. — 

The  groups  D3™  arise  by  reflecting  groups  D™  in  a  horizontal  plane  which 
either  contains  the  two-fold  axes,  (Sn),  or  lies  midway  between  them,  (So,). 


205.  D3'k={DLSh] 

206.  D3l.=  {D^S„, 

207.  D3l={DLS,l 

208.  D3l.=  {DLS^ 


Hexagonal  ietartohedry. — 

The  space-groups  isomorphous  with  Cg  result  from  combining  a  six-fold 
screw-axis  with  the  hexagonal  lattice. 


209. 

CJ  = 

M3>4- 

210. 

ci= 

K3't)4 

211. 

ci= 

r\3'  Ty'  ^T 

212. 

ct- 

K3.|').4- 

213. 

cl= 

Ka-T').-}- 

214. 

c?= 

Ki  n).  4. 

38  HEXAGONAL   SPACE-GROUPS. 

Hemimorphic  hemihedry. — 

The  groups  C^  are  obtained  by  combining  groups  C"  with  the  operation 
of  a  vertical  reflecting  plane  which  passes  through  either  the  line  AA'  or  the 
line  AB  of  figure  26.  The  reflection  in  the  plane  through  AA'  will  be  desig- 
nated as  Sft. 

215.   C6V={cS,sj.       r, 

216.  Cel={a,S,(rJ}.     T, 

217.  Ce^={C«,SJ.  r^ 

218.  Ce*.=  {C«,S,(r.)}.     T, 

Pammorphic  hemihedry, — 

The  space-groups  isomorphous  with  Cg  can  be  obtained  by  reflecting  groups 
C  ™  in  a  horizontal  plane. 

219.  C6\,=  {CJ,S,}.    r^ 

220.  c=L={c«,s^}.   r, 

Enantiomorphic  hemihedry. — 

The  space-groups  D™  are  most  simply  derived  by  combining  groups  C" 
with  the  operation  of  a  two-fold  axis  which  coincides  with  the  X-axis  of  coor- 
dinates of  the  point  and  isomorphous  space-groups  (AA'  of  figure  26).  This 
two-fold  rotation  will  be  represented  by  U». 

221.  DJ={CJ,UJ.  r^ 

222.  D^  =  {q,Ua}.  r,. 

223.  D^  =  {C^,Ua}.  r^ 

224.  D^={C*,  UJ.  r^ 

225.  Di={C6^Ua}.  r^  ' 

226.  D«={C^,UJ.  r^ 

Holohedry. — 

The  groups  Dg™  result  by  combining  groups  D^  with  an  inversion  which 
lies  in  the  six-fold  axis  either  at  its  intersections  with  the  two-fold  axes  (I) 
or  at  points  midway  between  such  intersections,  (I'). 


227. 

DeV'iDj,!}. 

r». 

228. 

De1.=  {Dj,I'}. 

T^ 

229. 

D/.  =  {D«,I}. 

Th 

230. 

DA  =  {D«.I'}. 

r. 

CHAPTER  III. 
THE  APPLICATION  OF  THE  THEORY  OF  SPACE- 
GROUPS  TO  CRYSTALS.* 
UNITS  OF  STRUCTURE. 

A  space  lattice  has  been  definedf  as  the  sum  total  of  the  points  of  inter- 
section of  any  three  sets  of  planes.  These  sets  of  planes  partition  the  space 
into  units  of  structure,  all  of  the  same  size  and  shape.  Such  a  unit  is 
OABDEGFC  of  figure  18.  There  will  thus  be  a  unit  corresponding  to  each 
of  the  14  lattices;  points  of  the  lattice  will  be  found  at  each  of  the  corners  of 
the  unit  prisms  and  in  some  cases  other  points  of  the  lattice  will  lie  in  the 
center  of  the  unit  or  at  the  centers  of  faces  (as  examples,  To'"  and  To").  If 
the  lattice  is  a  monocUnic  lattice,  the  unit  will  be  some  sort  of  a  monoclinic 
prism;  if  the  lattice  is  cubic,  the  unit  will  be  a  cube,  and  so  on. 


/ 


Fia.  27.  The  unit  cell  derived  from  Ttr.  The 
edges  of  this  unit  are  of  unequal  lengths  and 
make  unequal  angles  with  one  another. 

Just  as  a  simple  lattice  can  be  divided  intd  unit  prisms  by  three  sets  of 
planes  parallel  to  the  axes  of  coordinates,  so  any  space  grouping  of  points, 
built  upon  some  lattice,  can  be  similarly  divided.  The  fourteen  units  of 
structure  characteristic  of  the  fourteen  space  lattices  are  shown  in  figures 
27  to  34.  The  number  of  the  points  of  the  lattice  to  be  associated  with  a 
unit  prism  can  be  readily  told.  For  instance,  in  the  case  of  the  simple  cubic 
lattice,  Fc,  this  number  is  one  since  each  of  the  eight  points  of  the  lattice 
located  at  the  eight  corners  of  the  cube  is  shared  by  the  seven  other  cubes 
meeting  at  this  point  and  there  are  no  other  lattice  points  contained  in  or  touch- 
ing the  unit.  For  the  same  reason  the  unit  cube  of  a  space  grouping  having 
this  lattice  fundamental  to  it  will  have  a  single  group  of  equivalent  points 
(the  n  points  about  a  single  point  of  the  lattice)  associated  with  it;  each  of  the 
8  corner-points  of  the  lattice  will  contribute  to  the  cube  one  eighth,  and  each 
a  different  eighth,  of  the  equivalent  points  ranged  about  it, 

♦P.  Niggli,  op.  cit.;  Ralph  W.  G.  Wyckoff,  Am.  J.  Sci.  1,  127.     1921. 
t  See  p.  22. 

39 


40 


UNIT   OF   STRUCTURE   FOR   SPACE-GROUP   CL 


A  consideration  of  the  unit  of  the  space-group  aheady  discussed  in  detail, 
Cah,  will  make  this  more  clear.  The  unit  prism,  OAFCGBDE  of  figure  20 
(see  also  figure  18),  contains  four  equivalent  points  M,  M',  M",  and  M'",  the 
coordinates  of  which  are  M(xyz),  M'(2rx  — x,  2ry  — y,  z),  M"(x,  y,  2rz  — z)  and 
M'"(2rx— X,  2ry— y,  2rz  — z).  Since,  however,  the  arrangement  about  every 
point  of  the  lattice  is  the  same  as  that  about  every  other,  it  follows  that 
corresponding  points  of  the  groups  about  neighboring  points  of  the  lattice 
are  entirely  similar.  It  is,  then,  so  far  as  the  expression  of  the  relative  posi- 
tions of  equivalent  points  is  concerned,  permissible  to  consider  2rx— x=  —  x, 
—  y  =  2ry  — y,  and  —  z  =  2r^  — z.*  The  coordinates  of  the  four  equivalent 
positions  of  the  unit  of  structure  of  the  space-group  CgV  are  thus: 

xyz;     -X,  -y,  z;    x,  y,  -z;     -x,  -y,  -z 
or,  as  it  will  hereafter  be  written: 

xyz;    xyz;    xyz;    xyz. 
The  number  of  points  of  the  lattice  to  be  associated  with  the  units  of  each 
of  the  other  lattices  can  be  similarly  obtained  and  from  this  the  coordinates 

which  can  be  taken  as  typical  of  the  posi- 
tions of  equivalent  points  within  the  unit 
of  any  space-group  can  be  written  down. 
The  treatment  of  a  slightly  more  com- 
pUcated  space-group  will  outhne  the 
necessary  procedure.  For  this  purpose 
we  will  take  the  space-group  C2I1  obtained 
by  placing  the  point-group  C2  at  the 
points  of  the  second  monocHnic  lattice 
Fm  (figure  29).  The  unit  prism  of  this 
lattice  proves  to  be  a  monoclinic  prism 
with  additional  points  of  the  lattice  at  the 
centers  of  two  of  its  faces.  The  eight 
points  of  the  lattice  that  are  located  at 
the  corners  of  the  prism  serve,  as  with 
the  space-group  C2h,  to  place  within  it  the 
equivalent  points  of  one  group  (in  this 
instance,  by  definition,  a  point-group). 
One  half  of  the  points  about  each  of  the  two 
points  of  the  lattice  at  the  diagonals  of  faces  (and  opposite  halves)  he  within  the 
unit  prism  so  that  these  two  points  of  the  lattice  together  contrive  to  place  within 
the  unit  a  second  group  of  equivalent  points.     If  O  of  figure  29  is  taken  as 

*  This  simplification  is  geometrically  justified  (1)  since  the  unit  prism  that  has  been  chosen 
has  no  particular  physical  significance  but  serves  rather  as  a  unit  that  is  conveniently  visualized 
and  (2)  because  the  coordinates  adopted  actually  define  a  group  of  equivalent  points  which  re- 
peated along  and  parallel  to  the  axes  of  coordinates  will  build  up  the  entire  assemblage.  It  is, 
moreover,  justified  analytically  as  an  expression  of  the  points  associated  with  the  unit  prism  itself  (if 
one  prefers  to  think  of  this  unit)  because  as  applied  to  the  study  of  the  structure  of  crystals,  these 
coordinates  define  the  interference  effects  to  be  expected  from  atoms  placed  at  these  positions; 
this  definition  involves  sine  and  cosine  terms  within  which  2tx,  2Ty,  and  2tz  in  2tx— x,  etc.,  dis- 
appear. 


Fig.  28.  If  OX  j^OY  5^  YZ  and  ZY  is 
normal  to  the  plane  YOX  but 
ZYOXt^  90°,  this  is  the  unit  of 
TrnJ  if  the  three  edges  are  mu- 
tually perpendicular  and  (1) 
0X?^0Y?^YZ,  the  unit  corre- 
sponds to  To,  (2)  if  OX=OY?^YZ 
it  corresponds  to  rt(a)  or  (3)  if 
OX  =  0Y  =  YZ  the  unit  is  that  of  r.. 


THE  UNIT  OF  STRUCTURE  FOR  SPACE-GROUP  CI- 


41 


the  origin,  the  centers  of  the  second  group  of  equivalent  points  will  be  for  the 
half  of  the  equivalent  points  at  P(0,  Ty,  Tz)  and  for  the  other  half  at  the  oppo- 
site point  P'(2tx,  Ty,  Tz).     Keeping  in  mind  the  analogous  case  of  C2h  (figure 
20)  the  actual  coordinates  of  the  equivalent  points  within  this  unit  are:* 
xyz  ;    2rx-x,  2ry-y,  z      ;    x,  y,  2tz-z      ;    2rx-x,  2ry-y, 

2t,-z; 
X,  y+Ty,  z+Tz;    2rx-x,  Ty-y,  z+r^;    x,  y+Xy,  Tz-z;    2t^-x,  Ty-y, 

Tz— Z. 

Just  as  was  done  for  the  space-group  Cjh  these  coordinates  can  be  reduced  to : 
xyz  ;    xyz  ;    xyz  ;    xyz  ; 

x,  y+Ty,  z+Tzi    5c,  Ty-y,  z+Tz;    x,  y+Ty,  Tz-z;    x,  Ty-y,  Tz-z. 
It  will  be  observed  that  this  process  is  equivalent  to  placing  a  group  of  equiva- 
lent points  (in  this  case  a  point-group)  at  the  origin  and  at  one  other  point 

(0,     Ty,     Tz). 


Fig.  29.  If  YZ  J_  plane  YOX  and  ZY?^ 
YO  5^  OX  and  (1)  if  Z  YOX  5^  90°, 
the  unit  corresponds  to  Tm',  (2)  if 
Z  YOX =90°,  it  corresponds  to 
To'  (b). 


Fig.  30.  This  unit  is  a  rectangular 
parallelopiped;  if  YZp^YO^^OX 
it  corresponds  to  To' (a),  if^ZY 
5^Y0=0X  to  rt(b). 


The  positions  of  the  equivalent  points  within  a  unit  for  each  of  the  space- 
gi'oups  can  be  expressed  in  the  same  way  as  the  coordinates  of  the  character- 
istic groups  of  equivalent  points  placed  at  typical  points  of  the  lattice. f  The 
typical  point  or  points  of  the  lattice  corresponding  to  a  particular  unit  are  in 
all  cases  the  origin,  as  well  as  sometimes  the  center  of  the  unit  or,  as  in  this 
latter  instance,  Cah,  the  center  of  a  side  or  the  centers  of  several  sides.  The 
extension  of  this  same  line  of  thought  to  the  rest  of  the  14  lattices  will  show  the 
number  of  groups  of  equivalent  points  to  be  associated  with  the  unit.  Thus 
the  coordinates  of  typical  points  of  the  lattice  which  serve  as  centers  of  these 
groups  are  those  of  Table  2. 

*  This  is  true  if  x  is  less  than  tx,  y  than  Ty  and  z  than  tz.  A  slight  and  obvious  modification 
which  would  yield  final  and  reduced  values  the  same  as  these,  would  define  the  points  within  this 
unit  prism  if  one  or  all  of  x,  y  and  2  exceed  tx,  ry  or  t,. 

t  The  general  case  of  each  space-group  (Chapter  IV)  in  which  there  are  three  vaiiable  par^ 
meters  is  obtained  by  placing  the  characteristic  group  of  equivalent  points  at  the  typical  points 
of  the  underlying  lattice. 


42 


THE    14   UNITS   OF   STRUCTURE. 


Table  2. 

Number  of 

Lattice.           associated 

Coordinates  of 

lattice  points. 

typical  points. 

Triclinic  System. 

1.          Ttr 

1 

0  (COO). 

Fig.  27. 

MoNocLiNic  System. 

2.     r„, 

1 

0  (COO). 

Fig.  28. 

3.     r^' 

2 

O(CCO);  P(0,  ry,r.).      " 

Fig.  29. 

Orthorhombic  System 

4.      To 

1 

0  (COO). 

Fig.  28. 

5a.    To'  (a) 

2 

O(OOO);  Pi(T.,ry,0). 

Fig.  30. 

b.    To'  (b) 

2 

O(CCC);  P(0,Ty,r.). 

Fig.  29. 

6.      To" 

4 

O(CCC);P(0,ry,r.); 

Pi(Tx,Ty,C);  P2(r.,0,r.). 

Fig.  31. 

7.      To"' 

2 

O(CCC);  Pa  (tx,  Ty,  r.). 

Fig.  32. 

Tetragonal  System. 

8a.    Ft  (a) 

1 

0  (CCO). 

Fig.  28. 

b.    Ft  (b) 

2 

O(CCC);  P,  {T.,Ty,0). 

Fig.  30. 

9a.    Ft'  (a) 

4 

O(CC0);P(0,r„T.); 

Pi(rx,ry,0);  P2  (r.,  0,  r.). 

Fig.  31. 

b.    T,'  (b) 

2 

O(OGO);  P3  (tx,  Ty,  T.). 

Fig.  32. 

Cubic  System. 

10.      Tc 

1 

0  (COO). 

Fig.  28. 

11.     r/ 

4 

O(OOO);  P(0,Ty,rO; 

Pi(rx,ry,0);  P2  (r.,  0,  r,). 

Fig.  31. 

12.      To" 

2 

O(000);P3(tx,  Ty,  r.). 

Fig.  32. 

Hexagonal  System.* 

13.           Trh 

1 

0  (CCO). 

Fig.  33. 

14.      Th 

1 

0  (CCO). 

Fig.  34, 

SPACE-GROUPS  AND  CRYSTALS. 
Every  crystal,  considered  as  a  regular  arrangement  of  atoms  in  space, 
must  possess  the  symmetry  of  some  one  of  the  230  space-groups.  The  theory 
of  space-groups,  then,  supplies  a  method  with  the  aid  of  which  it  should  be 
possible  to  represent  all  of  the  ways  in  which  the  atoms  of  a  crystal  can  be 
arranged  in  space.  If  an  atom  of  a  crystal  occupies  such  a  position  that  it 
corresponds  with  the  coordinate  position  xyz  of  an  equivalent  point  of  the 
space-group  having  the  symmetry  of  the  crystal,  then  symmetry  demands  that 
exactly  similar  atoms  shall  be  found  at  positions  corresponding  to  those  of 

*  The  unit  cell  for  Fh  can  also  Tbe  taken  as  a  base-centered  rhombic  prism,  the  lengths  of  whose 
sides  stand  in  the  ratio  of 

a  :  b  :  c  =  \/3  :  1  :  c. 
Niggli   (op.  cit.)  has,  worked  out  upon  this  basis  the  analytical  expression  for  all  of  the  groups 
having  Th  as  the  fundamental  lattice.     Such  a  unit  is  useful  when  it  is  desired  to  compare  an 
hexagonal  crystal  with  one  exhibiting  rhombic,  tetragonal  or  cubic  symmetry. 


SPACE-GROUPS   AND    CRYSTALS. 


43 


each  of  the  other  equivalent  points  of  the  space-group.  Most  crystals  are 
built  of  atoms  of  more  than  one  sort.  As  a  consequence  if  we  find  the  atoms 
of  kind  A  occupying  the  positions  of  equivalent  point  xyz  and  the  other  points 
equivalent  to  it,  the  atoms  of  B  will  be  found  at  some  other  positions  developed 
from  x'  y'  z',  and  so  on. 

The  atoms  of  a  crystal  may  thus  be  thought  of  as  occupying  the  positions 
of  a  sort  of  composite  space-group  developed  by  superimposing  several  sets 
of  equivalent  positions  upon  the  same  set  of  axes  (and  other  elements  of  sym- 
metry). The  atom.s  of  a  crystal,  as  a  result,  must  be  arranged  in  groups  with 
centers  at  the  points  of  one  of  the  space  lattices.     Such  a  group  of  atoms  has 


Fig.  31. 
=  0X 


A  rectangular  parallelopiped.  If 
YOt^OX  it  corresponds  to  To",  if 
YO=OX  to  rt'(a),  or  if  YZ  =  YO 

to  Tc'. 


Fig.  32.  A  rectangular  parallelopiped.  If 
YZ9^Y0  9^0X  it  corresponds  to  To'", 
if  YZ?^YO=OX  to  rt'(b),  or  if  YZ  = 
YO=OXtorc". 


been  called  a  crystal  molecule.  In  this  sense  the  crystal  molecule  is  a  purely 
geometrical  conception  and  except  under  special  conditions  would  not  be 
thought  of  as  possessing  any  physical  significance. 

It  is  possible,  of  course,  to  think  of  a  crystal  as  divided,  in  the  same  way 
that  a  space-group  can  be  divided,  into  a  large  number  of  unit  prisms  by  sets 
of  planes  passing  parallel  to  the  three  planes  each  of  which  contains  two  of 
the  axes  of  coordinates.  Measurements  of  the  X-ray  spectrum  from  the  face 
of  a  crystal  together  with  a  knowledge  of  the  density  of  the  crystal  can  be 
made  to  yield  the  nmnber  of  chemical  molecules  that  are  to  be  associated 
with  this  unit  of  structure.*  If  a  compound  were  of  the  type  AB,  where  A 
is  one  kind  of  atom  and  B  another,  and  if  the  atoms  of  A  occupy  the  most 
general  equivalent  positions  one  of  which  is  xyz,  then  there  will  be  as  many 
chemical  molecules  of  AB  associated  with  the  unit  prism  as  there  are  equiva- 

*  The  factor  actually  determined  is  n^/m,  where  n  is  the  "order"  of  the  reflection  spectium 
and  m  is  the  number  of  chemical  molecules  associated  with  the  unit  prism.  The  value  of  n  cannot, 
however,  in  general  be  determined  so  that  to  may  usually  have  one  of  two  or  perhaps  three  values. 


44 


SPECIAL   CASES   OF   SPACE-GROUPS. 


lent  points  in  the  unit.  This  number  may  under  certain  conditions  be  rela- 
tively great.  For  instance,  in  the  case  of  the  space-groups  having  the  sym- 
metry of  the  holohcdry  of  the  cubic  system,  the  number  of  equivalent  points  of 
the  point-group  0**,  and  of  the  other  groups  of  points  associated  with  a 
single  point  of  the  lattice,  is  48.  If  then  the  fundamental  lattice  of  a  holo- 
hedral  cubic  space-group  is  the  simple  cubic  lattice  Fc  and  the  compound 
crystalhzes  with  this  symmetry  (as  sodium  chloride  does,  for  instance),  48 
(if  all  of  the  A  atoms  are  alike  and  all  of  the  B  atoms  are  also  alike,  and 
more  if  they  are  not  alike)  chemical  molecules  of  AB  must  be  placed  within 
the  unit  cell;  if  the  lattice  were,  on  the  other  hand,  the  face-centered  lattice 
Fc  with  four  points  of  the  lattice  associated  with  the  unit,  this  number  of 
molecules  of  AB  must  be  at  least  192. 


Fig.  33.  If  the  three  edges  meeting  at  O 
are  of  equal  legths  and  make  equal, 
angles  with  one  another,  this  unit  cor- 
responds to  Tth. 


Fig.  34.  If  ZO  i  plane  YOX  and 
ZYOX  =  120°,  a  rhombic  prism 
two  of  the  sides  of  whose  base 
are  XO  and  OY  and  of  height 
OZ  serves  as  the  unit  for  Th. 


SPECIAL  CASES. 

If,  however,  the  values  of  x,  y  and  z  which  express  the  positions  of  the  atoms 
of  A  and  B  are  such  that  the  atoms  lie  upon  some  element  of  symmetry,  two 
or  more  of  the  equivalent  positions  coincide  and  this  number  of  molecules  to 
be  placed  within  the  unit  cell  will  be  reduced.  For  instance  if  a  point  were  to 
lie  upon  a  plane  of  symmetry,  it  would  of  course  be  identical  with  its  mirror 
image;  or  if  it  stood  in  a  three-fold  or  four-fold  axis  of  symmetry,  three  or 
four  of  the  equivalent  points  would  occupy  the  same  position.  In  the  space- 
group  C21J  (figure  20)  if  z  is  equal  to  t«,  that  is,  to  one  half  of  the  height  of 
the  unit  prism,  then  the  four  equivalent  points  of  the  unit  would  occupy  two 
positions  (M  coincides  with  M"  and  M'  with  M'")  or  if  x  is  equal  to  r,,  and  y 
to  Ty,  the  four  points  will  have  two  equivalent  positions  (M  will  coincide 
with  M'  and  M''  with  M'") .    Ifx=y  =  z=0  then  the  four  points  will  all  unite 


THE  TYPICAL  CASE  OF  CALCITE.  45 

at  the  origin  and  there  will  be  but  one  equivalent  position  within  the  unit; 
the  same  is  true  if  x  =  rx,  y=ry  and  z=Tj.. 

The  results  of  all  of  the  X-ray  experimentation  which  has  thus  far  been 
carried  out  seem  to  point  to  the  fact  that  this  number  of  chemical  molecules 
to  be  contained  within  a  unit  cell  is  in  all  probabmty  very  much  less  than  the 
number  of  most  generally  placed  equivalent  positions.  As  a  consequence  the 
determination  of  these  special  cases  of  the  space-groups  becomes  of  the  utmost 
importance  to  the  person  interested  in  the  structure  of  crystals. 

A  discussion  of  calcite,  which  has  already  been  treated  in  detail  by  this 
procedure,*  will  serve  to  indicate  the  need  for  these  special  cases  of  the  space- 
groups.  The  X-ray  measurements  show  that  almost  certainly  two  chemical 
molecules  of  calcium  carbonate  are  to  be  associated  with  a  unit  rhombohedron. 
Calcite  crystallizes  with  a  symmetry  which  is  that  of  the  point-group  D3. 
Two  space-groups  isomorphous  with  D3,  namely  Dg^  and  Dgd,  have  Trh  as 
the  fundamental  lattice.  Since  two  chemical  molecules  of  calcium  carbonate 
are  to  be  associated  with  the  unit  rhombohedron,  two  calcium  atoms,  two 
carbon  atoms  and  six  oxygen  atoms  must  be  placed  within  it.  These  two 
calcium  atoms  may  conceivably  be  alike  or  they  may  be  different  one  from  the 
other;  the  same  is  true  for  the  two  carbon  atoms;  and  the  oxygen  atoms  may 
be  for  instance  (1)  all  ahlce,  (2)  all  different,  (3)  four  ahke  and  two  different, 
(4)  two  sets  of  three  hke  atoms  or  (5)  three  sets  of  two  Uke  atoms.  Copjdng 
from  page  157  it  is  seen  that  all  of  the  potential  atomic  positions  consistent 
with  the  space  groups  Ds^  and  Y>^^  are 

Space-Group  Dgdi 

Oitie  equivalent  position: 

(a)  0  0  0.  (b)  H  i. 

Two  equivalent  positions: 

(c)  uuu;    tiuu. 
Three  equivalent  positions: 

(d)  OOi;    OiO;    ^00.  (e)  OH;    HO;     |0i 
&ix  equivalent  positions: 

(f)  utiO;    uOu;    Ouu;    uuO;    uOu;    Ouu. 

(g)  uu|;    u|u;    ^uti;    uu^;    u^ti;    |uu. 
(h)  uuv;    uvu;    vuu;    uuv;    uvti;    viiu. 

Twelve  equivalent  positions : 

(i)    xyz;    yzx;     zxy;    yxz;    xzy;    zyx; 
xyz;    yzx;    zxy;    yxz;    xzy;    zyx. 

Space-Group  Dg^: 

Two  equivalent  positions : 

(a)  000;    Hi  (b)  Hi;    f  f  f . 

Four  equivalent  positions : 

(c)  uuu;    tiuu;    ^-u,  |-u,  |-u;    u-j-|,  u-{-i  u-|-i 


*  Ralph  W.  G.  Wyckoff,  Am.  J.  Sci.  50,  317.     1920. 


46  THE   TYPICAL    CASE    OF   CALCITE. 

Six  equivalent  positions : 

{A\       111.       111.       113.       131.       111.        Ill 

Vu;        444>        4441        444>        444>        444?        444' 

(e)  uuO;    uOu;    Ouu;    |-u,  u+^,  |;    u+|,  |,  |-u; 

2>     2        U,    11+2- 

TweZi'g  equivalent  positions: 

(f)  xyz;     yzx;     zxy;  yxz;     xzy;     zyx; 

2     X,  2     Y)  2~2;;     2-     y,  2     ^^  2     ^j     2     z,  2     x,  2     yj 
y+i  x+i  z+i;     x+i  z+i  y+^;     z+i  y+i  x+|. 
The  attempt  to  write  down  on  the  basis  of  these  coordinate  positions  the 
different  arrangements  of  the  atoms  in  calcite  that  are  possible  in  the  light  of 
its  symmetry  immediately  eliminates  many  of  the  possibihties  just  discussed. 
For  instance  it  is  clear  that  in  neither  case  are  there  enough  special  cases  of 
one  equivalent  position  so  that  the  two  calcium  atoms  can  be  different  and 
the  two  carbon  atoms  also  different.     The  same  fact  shows  that  possibility 
(2)  for  the  arrangement  of  the  oxygen  atoms  may  also  be  omitted  from  con- 
sideration; it  can  be  similarly  shown  that  there  are  in  neither  space-group 
sufficient  special  cases  so  that  four  of  the  oxygen  atoms  can  be  alike  and  two 
dift'erent.     All  of  the  possible  ways  for  the  atoms  of  calcite  to  be  arranged 
can  then  be  written  as:* 
Arrangements  arising  from  Dg^: 

(a)  Ca  =  u  u  u ;    u  u  u. 

C  =  Ui  U]  ui ;    til  til  ti2. 

O  =  U2ti2  0;    ti2  0u2;    Ou2ti2;    112  U2O;    U2OU2;    0ii2U2. 

(b)  Ca  and  C  as  in  (a). 

O  =  U2 112 1;    112 1 U2;    §  U2 112;    ti2  U2 1;    U2  ^  112;    I  U2  U2 

(c)  Ca  and  C  as  in  (a). 

0  =  U2  U2  v;     U2  V  U2;     V  U2  U2;     tia  ti2  v;     112  v  112;     v  ti2  ti2. 

(d)  Ca  and  C  as  in  (a). 

0=U2U2U2;    ti2ti2ti2.     U3U3U3;    tiaiistis.     U4U4U4;    ii4  ti4  ti4. 

(e)  Ca  and  C  as  in  (a). 

0  =  00^;    0  10;    1 0  0.    OH;    HO;    |0i 
Arrangements  arising  from  DaV 

(f)  Ca  =  iH;    HI     or    000;     Hi 

C=000;    H^     or    iii;    f  f  f . 

0—133.        331.       111.       111.        311.        Ill 
~444;        444>        444>        444;        4   4   4>        4   44- 

(g)  Ca  and  C  as  in  (f). 

0  =  uuO;    tiOu;    Ouu;    ^-u,  u-Hi  ^;    u+^,  |,  i-u; 

2>     2~U,    U+2. 

In  this  same  manner  all  of  the  ways  of  arranging  the  atoms  in  any  crystal 
can  be  written  down  from  a  knowledge  of  the  number  of  molecules  to  be 
associated  with  the  unit  cell  (as  furnished  by  the  X-ray  spectrum  measure- 
ments) and  from  a  consideration  of  the  special  cases  of  the  different  space- 
groups  possessing  the  symmetry  of  the  crystal. 

*  These  arrangements,  giving  as  we  have  seen  the  positions  of  the  atoms  within  a  unit  cell 
which  by  simple  translations  along  the  axes  of  refeience  will  locate  all  of  the  atoms  in  the  ciystal 
are  in  a  form  which  is  immediately  usable  for  testing  them  by  fuither  X-ray  measurements. 


CHAPTER  IV. 

THE  COMPLETE  ANALYTICAL  EXPRESSION  OF 

THE  SPACE-GROUPS. 

Niggli  has  already  recorded  many  of  the  simpler  cases  for  the  various  space- 
groups.  For  some  time  the  present  writer  has  been  engaged  in  working  out 
analytically  all  of  the  special  cases  of  the  space-groups.  The  tables  which 
follow  are  the  results  of  these  computations.  They  purport  to  give  the  coordi- 
nates of  the  most  generally  placed  equivalent  points  and  all  of  the  special 
cases  of  these  equivalent  points  contained  within  the  unit  of  structure  of  each 
of  the  230  space-groups. 

The  analytical  determination  of  the  special  cases  can  be  quite  simply 
carried  out  by  equating  the  coordinates  of  one  point  xyz  with  those  of  each  of 
the  other  equivalent  positions  within  the  unit  cell.  This  will  yield  a  series 
of  special  cases  (if  any  exist)  which  can  be  further  speciaHzed  by  applying  this 
same  process  to  the  coordinates  of  these  special  positions.  The  continued 
use  of  this  procedure  will  eventually  yield  all  of  the  special  cases  for  a  space- 
group.*  By  way  of  illustration  the  special  cases  of  the  space-group  C2h 
(page  49)  will  be  deduced.  The  positions  of  the  most  generally  placed 
equivalent  points  in  the  unit  cell  of  this  space-group  are 
xyz;    xyz;    xyz;    xyz. 

Equivalent  point  xyz  will  have  the  same  position  as  equivalent  point  xyz 
when 

(1)  x=x,  y  =  y,  z  =  z;  that  is,  when  x=0  or  |  (Xa),  y  =  0  or  -|  (Xb)  and 
z=w(Xc)  where  w  is  any  fractional  part  of  c.  The  lengths  a,  b,  c 
are  unit  lengths  along  the  X-,  Y-  and  Z-axes. 

It  will  have  the  same  position  as  the  point  xyz  when 

(2)  x  =  x,  y  =  y,  z  =  z;  that  is,  when  x=u(Xa),  y  =  v(Xb),  z  =  0  or 

^(Xc);  u  and  v  are  any  fractional  parts  of  a  and  b,  respectively. 
The  points  xyz  and  xyz  will  coincide  in  position  when 

(3)  x  =  x,  y  =  y,  z  =  z;  that  is,  when  x  =  0  or  |(Xa),  y  =  0  or  |(Xb), 

z  =  Oor|(Xc). 
The  special  cases  of  this  space-group  then  arise  from  using  these  values 
for  X,  y  and  z.     They  are 
From  (1) : 
(a)  when  x=0,  y=0,  and  z  =  w;t  then  OOw;    0  0  w. 

*  The  algebra  of  this  process  differs  in  certain  details  from  the  more  ordinary  kind.  For  in- 
stance there  arises  from  our  previous  definitions  the  fact  that  0  =  1=2= Further- 
more x=x=0  or  I,  and  X  =  K—  X  =  i  or  J,  and  more  generally  x  =  l/n  — x  =  "^"  ,  where  n  =  l, 
2,  3, 

fin  this  example  and  in  all  of  the  tables  which  follow  only  the  fractional  parts  of  the  unit 
lengths  along  the  diffeient  coordinate  axes  will  be  stated.  If  for  any  reason  absolute  distances 
of  points  are  desired,  it  is  of  course  necessary  to  multiply  the  coordinate  values  given  in  these 
tables  by  the  proper  values  of  a,  b  and  c. 

47 


48  THE   TRICLINIC   SPACE-GROUPS   c}    AND    cj. 

(b)  whenx  =  0,  y  =  |,  z=w;  then  0|w;  0  |  w. 

(c)  whenx  =  |,  y  =  0,  z  =  w;  then  |0w;  ^  0  w. 

(d)  whenx  =  f,  y  =  h  z  =  w;  then  H  w;  |  i  w. 
From  (2) : 

(e)  whenx  =  u,  y  =  v,  z  =  0;  then  uvO;  tivO. 

(f)  whenx  =  u,  y  =  v,  z  =  §;  then  uv|;  u  v  |. 

From  (3) : 

(g)  whenx  =  y  =  z=0;  then    0  0  0. 
(h)  whenx  =  ^,  y  =  z  =  0;  then    ^0  0. 
(i)    whenx  =  z  =  0,  y  =  ^-;  then    0^0. 
(j)   whenx  =  y  =  0,  z  =  ^;  then    0  0|. 


(k)  whenx=0,  yandz  =  |;  then     0| 


1  1 

2- 


(1)  when  x  and  z  =  |,  y  =  0;  then  ^  0  |. 
(m)  when  x  and  y  =  ^,  z  =  0;  then  ^  |  0. 
(n)  when  x,  y  and  z  =  | ;  then    §  1 1. 

We  must  now  speciaHze  by  the  same  procedure  each  of  the  special  cases 
(a)  to  (f).  Inspection,  however,  shows  that  in  the  present  instance  this  will 
lead  to  no  new  special  positions.  All  of  the  special  cases  of  the  space-group 
C2h  are  then  defined  by  (a)  to  (n). 

The  other  space-groups  can  all  be  specialized  in  the  same  fashion.     These 
special  positions  for  each  space-group  are  given  in  the  tables  which  follow. 

TRICLINIC  SYSTEM. 

A.  HEMIHEDRY. 
Space-Group  C}. 

One  equivalent  position: 
(a)  xyz. 

B.  HOLOHEDRY. 


Space-Group  C}. 

One  equivalent  position: 

(a)  0  0  0.  (e)  H  0. 

(b)OOi  (f)   |0|. 

(c)  0^0.  (g)OH. 

(d)iOO.  (h)HI. 

Two  equivalent  positions: 
(i)   xyz;    xyz. 


THE   MONOCLINIC   SPACE-GROUPS   Cg-Czh.  49 

MONOCLINIC  SYSTEM. 

A.  HEMIHEDRY. 
Space-Group  Cl. 

One  equivalent  position: 

(a)  u  V  0.  (b)  u  V  |. 

Two  equivalent  positions: 
(c)  xyz;    xyz. 

Space-Group  Cl. 

Two  equivalent  positions: 
(a)  xyz;    x+i  y,  z. 

Space-Group  Cf. 

Two  equivalent  positions : 

(a)  u  V  0;    u,  v-f  ^,  ^. 
Four  equivalent  positions : 

(b)  xyz;    xyz;    x,  y-j-f,  z+|;     x,  y+i  |-z. 

Space-Group  Cg. 

Four  equivalent  positions: 

(a)  xyz;    x-|-i  y,  z;    x,  y+|,  z-f^;    x-M,  y+|,  |-z, 

B.  HEMIMORPHY. 
Space-Group  Cl. 

One  equivalent  position: 

(a)  0  0  u.         (b)  I  0  u.         (c)  0 1  u.        (d)  H  u. 
Two  equivalent  positions : 
(e)  xyz;    xyz. 

Space-Group  Cg. 

Two  equivalent  positions: 
(a)  xyz;    x,  y,  z-f  ^ 

Space-Group  Cl. 

Two  equivalent  positions: 

(a)  OOu;    0,  I,  u-M.        (b)^Ou;    i  |,  u-Ff. 
Four  equivalent  positions: 

(e)  xyz;    xyz;    x,  y-f-i  z-{-^;    x,  ^-y,  z+^. 

C.  HOLOHEDRY. 
Space-Group  €2^. 

One  equivalent  position: 

(a)  0  0  0.  (e)  OH- 

(b)OOi  (f)   hO-l 

(c)iOO.  (g)HO. 

(d)OiO.  (h)Hi 


50  THE   MONOCLINIC   SPACE-GROUPS   C2h-C2l, 

Space-Group  C2h  {continued). 
Two  equivalent  positions: 


(i)    OOu;    OOu. 

(1)    Hu; 

HQ. 

(j)    0|u;    OiQ. 

(m)  u  vO; 

uvO. 

(k)  |0u;    |0u. 

(n)  uv|; 

U  V  |. 

Four  equivalent  positions : 

(o)  xyz;    xyz;    xyz; 

xyz. 

Space-Group  Cgh- 

Two  equivalent  positions: 

(a)  OOi;    oof. 

(d)Hi; 

113 

2    2    4- 

(b)0H;   OH. 

(e)  uvO; 

u  v|. 

(c)  hOh    |0f. 

Four  equivalent  positions : 

(f)    xyz;     X,  y,  z+|;     xyz;     x,  y,  ^-z. 
These  coordinate  positions  can  be  simplified  by  transferring  the  origin  to 

the  point  { i/  )  of  this  first  set.     They  then  become : 
Two  equivalent  positions: 


(a)  0  0  0; 

OOi. 

(d)HO; 

Hi 

(b)  0|0; 

OH. 

(e)  uvi; 

u  vf. 

(c)  |0  0; 

^Oi 

Four  equivalent  positions: 

(f)   xyz; 

X,  y,  z+l; 

X,  y,  |-z; 

xyz. 

Space-Group  C2I. 

Two  equivalent  positions : 

(a)  0  0  0; 

n  i  i 

u  2  2. 

(c)  10  0; 

Ill 
222. 

(b)OOi; 

0^0. 

(d)  HO; 

hoh 

Four  equivalent  positions: 

(e)  OH; 

nil.     0  3 

U  4  4  ,       U  4 

f;   Oil. 

(f)  244; 

13    1.        13 
2   4   4)        2    4 

3  .        113 
4)        244. 

(g)  OOu; 

OOti;    0, 

h  u+A;    0, 

1    1  _i, 

2)2         l^l. 

(h)  iOu; 

iOu;    h 

h,  u+l;    h 

i  Hu. 

(i)   uvO; 

uvO;    u, 

v+i  1;    u, 

2        V,     2" 

Eight  equivalent  positions: 

(J)   xyz; 

xyz;    x,  y-fi  z-|-|;    x, 

^-y,  z+i; 

xyz; 

xyz;    X,  ^- 

-y,  Hz;    X, 

y+h  h-z. 

Space-Group  C2l». 

Two  equivalent  positions: 

(a)  iOO; 

f  0  0. 

(d)HO 

;   HO. 

(b)  i  i  i; 

IH; 

(e)  OOu 

;    |0u. 

(c)  iO|; 

fO|. 

(f)   Hu 

;    0|u. 

THE   MONOCLINIC   SPACE-GROUPS   CztrCzh-  51 

Space-Group  C2h  {continued). 
Four  equivalent  positions : 

(g)  xyz;     xyz;     x+i  y,  z;     ^-x,  y,  z. 
These  coordinate  positions  can  be  simplified  by  transferring  the  origin  to 
the  point  (  ;c  )  of  this  first  set.     They  then  become : 

Two  equivalent  positions : 


(a)  0  0  0;    |0  0. 

(b)  0  II;    III. 

(c)  001;    lOi 

(d)0|0; 

(e)  fOu; 

(f)  flu; 

110. 
f  Ou. 
A  i  fi 

4    2   U. 

Four  equivalent  positions: 
(g)  xyz;    l-x,  y,  z; 

x+l,  y,  z; 

xyz. 

SpACE-Group  C2h. 

Two  equivalent  positions : 
(a)  iOi;    fOf. 

/'K^i    1  1  1  •        3   13 
\")    4   2   4>        4   2    4- 

(c)  iOf; 
(d)ilf; 

3  n  1 

4^4' 

3  11 

4  2   4- 

Four  equivalent  positions: 

(e)  xyz;    x,  y,  z+|;    x+|,  y,  z;    |-x,  y,  |-z. 
These  coordinate  positions  can  be  simplified  by  transferring  the  oiigin  to 
the  point  l-^,  -^ )  of  this  first  set.     They  then  become : 

Two  equivalent  positions : 

(a)  0  0  0;    1 0|.  (c)  0  0|;    |0  0; 

(b)0|0;    III.  (d)  Oil;    1 1 0. 

Four  equivalent  positions : 

(e)  xyz;    |-x,  y,  z+|;    x+|,  y,  |-z;    xyz. 
Space-Group  CgV 

Four  equivalent  positions : 


(a)  iOO; 

fOO; 

111. 

4    2    2> 

III. 

(b)iO|; 

fO|; 

1 1  n- 

4    2   »-'> 

f|0. 

(c)  Hi; 

3    3   1. 

3   13. 

Ill 

4   4   4; 

4    4    4> 

444. 

(d)fff; 

111. 

111. 

111 

4  4   4; 

4    4    4> 

444. 

(e)  OOu; 

|0u; 

0,  1, 

u+l;    1,  1,  1 

..     _.  _.  t-u. 
Eight  equivalent  positions: 

(f)   xyz;    xyz;    x+|,  y,  z;    |-x,  y,  z; 

X,  y+l,  z+^;    X,  |-y,  z+l;    x+i  y+i  |-z; 

2~X,     2~y>     2~Z. 

A  change  of  origin  to  the  point  (  ;^  )  of  this  set  of  coordinates  would  simplify 
(a)  and  (b). 


62  THE    ORTHORHOMBIC   SPACE-GROUPS   Cav-Ca^y 

ORTHORHOMBIC  SYSTEM. 

A.  HEMIMORPHY. 
Space-Group  CgV. 

One  equivalent  position: 

(a)  OOu.  (c)  iOu. 

(b)  0  I  u.  (d)  H  u. 
Two  equivalent  positions : 

(e)  uOv;    tiOv.  (g)  Ouv;    0  u  v. 

(f)  u|v;    u|v.  (h)  |uv;    ^  u  v. 

Four  equivalent  positions : 

(i)    xyz;    xyz;    xyz;    xyz. 
Space-Group  Ciy. 

Two  equivalent  positions : 

(a)  uOv;    u,  0,  v-f-|. 
Four  equivalent  positions : 

(c)  xyz;    X,  y,  z+|;    xyz; 
Space-Group  Cav- 

Two  equivalent  positions: 

(a)  OOu;    0,  0,  u+|. 

(b)  O^u;    0,  h  u+i 

Four  equivalent  positions : 

(e)  xyz;    xyz;    x,  y,  z+^;    x,  y,  z+|. 

Space-Group  Cgtr. 

Two  equivalent  positions : 

(a)  OOu;    ^u.        (b)  Hu;    0|u.        (c)  iuv;    fuv. 
Four  equivalent  positions: 

(d)  xyz;    xyz;    x-|-|,  y,  z;    ^-x,  y,  z. 

Space-Group  Cgy. 

Four  equivalent  positions: 

(a)  xyz;    x,  y,  z-f-|;    x+i,  y,  z;    ^-x,  y,  z-\-^. 

Space-Group  CaV 

Two  equivalent  positions: 

(a)  OOu;    i  0,  u+|.  (b)Hu;    0,  i  u+|. 

Four  equivalent  positions: 

(c)  xyz;    xyz;    x+i  y,  z-\-^;    |-x,  y,  z-|-i 


(b) 

u|v; 

u, 

i 

v+i 

X,  ; 

y,  z+i 

(c) 

iOu; 

h 

0, 

u-l-i 

(d) 

Hu; 

1 

2> 

1 

2"> 

u+|. 

THE   ORTKOREOMBIC   SPACE-GROUPS   Cay-Cay.  53 

Space-Group  Cay. 

Two  equivalent  positions: 

(a)  |uv;    f,  u,  v+§. 

Four  equivalent  positions: 

(b)  xyz;    x,  y,  z+|;    x+i  y,  z+l;    |-x,  y,  z. 

A  slight  simplification  can  be  effected  by  transferring  the  origin  of  coordi- 
nates  to  -~  of  this  first  set.     They  then  become : 

Two  equivalent  positions : 

(a)  Ouv;    I,  u,  v+|. 
Four  equivalent  positions: 

(b)  xyz;    |-x,  y,  z+|;    x+i  y,  z+i;    xyz. 

Space-Group  CgV 

Two  equivalent  positions: 

(a)  OOu;    Hu.  (b)  ^Ou;    0 1  u. 

Four  equivalent  positions: 

(c)  xyz;    xyz;    x-|-|,  |-y,  z;    ^-x,  y+|,  z. 

Space-Group  Cay. 

Fowr  equivalent  positions: 

(a)  xyz;    x,  y,  z+|;    x-fi  |-y,  z;    |-x,  y-^i,  z-{-i 

Space-Group  Cgy. 

Tioo  equivalent  positions: 

(a)  OOu;    i  i  u-hi        (b)Oiu;    I,  0,  u-f^. 

Fowr  equivalent  positions : 

(c)  xyz;    xyz;    x+i  |-y,  z+i;    ^-x,  y+|,  z+|. 

Space-Group  CH. 

Two  equivalent  positions: 

(a)  OOu;    Hu.  (b)  ^Ou;    0  §  u. 

Four  equivalent  positions : 

(c)  Hu;    f  fu;    Hu;  f  iu. 

(d)uOv;    uOv;    u-}-|,  |,  v;  ^-u,  |,  v. 

(e)  Ouv;    Ouv;    ^,  u+^,  v;  i,  ^-u,  v. 

iJzg/i<  equivalent  positions: 

(f)  xyz;  xyz;  xyz;  xyz; 

x+i  y+iz;    l-x,  §-y,  z;    x-|-i  ^-y,  z;    |-x,  y-j-|,  z. 


54  THE    ORTHORHOMBIC   SPACE-GROUPS   cH-cl 

Space-Group  Cl^. 

Four  equivalent  positions: 

(a)  uOv;    fi,  0,  v+|;    u-I-^  §,  v;    |-u,  i  v+ 


1 

2- 

Eight  equivalent  positions: 

(b)  xyz;  x,  y,  z+l;  xyz;  x,  y,  z+i; 
x+i  y+l,  z;    ^-x,  |-y,  z+l;    x+|,  |-y,  z; 

i-x,  y+i  z+|. 
Space-Group  Cgv- 

/^owr  equivalent  positions : 

(a)  OOu;    Hu;    0,  0,  u+i;    i  i  u+i. 
(b)Oiu;    lOu;    0,  i,  u+A;    i  0,  u+i. 

(c)  4  4  u;     4  4  u;     4,  4,  u+fJ     4>  4>  u-|-2. 
Efy/i/  equivalent  positions : 

(d)  xyz;  xyz;  x,  y,  z+|;  x,  y,  z+^; 
x+^y+iz;    l-x,  ^-y,  z;    x+|,  |-y,  z+|; 

i-x,  y+i  z+|. 
Space-Group  Cgv- 

Two  equivalent  positions : 

(a)  OOu;    0,  i  u+|.  (b)  H  u;    |,  0,  u+i 

Four  equivalent  positions: 

(c)  uOv;    uOv;    u,  |,  v-j-|;  u,  I,  v+|. 

(d)Ouv;    Ouv;    0,  u+|,  v+i;    0,  |-u,  v+|. 

(e)  |uv;    |uv;    i  u-M,  v+|;    i  |-u,  v+|. 

^^gf/if  equivalent  positions: 

(f)  xyz;  xyz;  xyz;  xyz; 

X,  y+l,  z+^;     X,  ^-y,  z-i-l;     x,  |-y,  z+§;    x,  y-|-|,  z+|. 

Space-Group  Czv- 

Four  equivalent  positions : 

(a)  OOu;    O^u;    0,  0,  u4-|;     0,  i  u-}-|. 
(b)Hu;    |0u;    hh^+i;     I,  0,  u-f-|. 

(c)  uiv;    uf  v;    u,  f,  v-|-|;     u,  i,  v-f-|. 

Eight  equivalent  positions: 

(d)  xyz;  syz;  x,  y,  z+|;    x,  y,  z-f-J; 
X,  y+iz+l;    X,  ^-y,  z4-i;    x,  |-y,  z;    x,  y+l,  z. 

Space-Group  Cay. 

Four  equivalent  positions : 

(a)  OOu;    |0u;    0,  |,  u+i;  h  h  u+i 

(b)  iuv;    fuv;    i  u-l-^  v+^;    f,  ^-u,  v+i 


THE   ORTHORHOMBIC   SPACE-GROUPS   cl^-cly.  55 

Space-Group  C2V  (continued). 
Eight  equivalent  positions : 

(c)  xyz;  xyz;  x+|,  y,  z;  ^-x,  y,  z; 

X,  y+^z-l-^;    X,  ^-y,  zH-l;    x+|,  |-y,  z-hl; 

i-x,  y-hi  z-hi. 
Space-Group  CH. 

Four  equivalent  positions : 

(a)  OOu;    iiu;    |,  0,  u+i;    0,  i,  u-j-i 
Eight  equivalent  positions: 

(b)  xyz;  xyz;  x+|,  y,  z+i;    ^-x,  y,  z+|; 
X,  y+iz+^;    X,  i-y,  z+l;    x+i  ^-y,  z;     |-x,  y+^,  z. 

Space-Group  CH. 

Four  equivalent  positions: 

(a)  OOu;    Hu;    i  0,  u-h|;    0,  i  u-|-i 
Eight  equivalent  positions: 

(b)  4  4U;     4  4U;     T)  T>  U"r2;     4>  4j  u~r2J 
13„.     3i„.     13    ,,4.1.     3     1    n_l_l 

4   4  U,       4   4  U,       4,    4,    UT^2;       4j    4)    '^T^2' 

(c)  uOv;  uOv;  u+|,  i  v;    |-u,  I,  v; 
u+i,  0,  v+§;    |-u,  0,  v-(-i;    u,  ^,  v-H|;    u,  |,  v-|-|. 

(d)  Ouv;  Ouv;  i  u-hl,  v;  i  |-u,  v; 
iu,  v-hl;    ^,  u,  v-h^;    0,  u-M,  v-|-|;    0,  ^-u,  v+|. 

Sixteen  equivalent  positions : 

(e)  xyz;  xyz;  xyz;  xyz; 

x+2)  y~r2>  z;  2~x,  ^— y,  z;  x-|-2,  2"~y>  z;  2~x,  yi-2>  z; 

x+iy,  z-M;  ^-x,  y,  z+l;  x+i  y,  z-|-^;  |-x,  y,  z-|-|; 

X,  y+iz-f-|;  X,  i-y,  z-l-i;  x,  f-y,  z-l-|;  x,  y-|-|,  z-hi 

Space-Group  C2V. 

Eight  equivalent  positions: 

(a)  OOu;  Hu;  i  i  u+i;    f,  f,  u-hi; 
iO,u+i;    0,iu-F^;    f,  i  u-^f;    i  f,  u-|-|. 

Sixteen  equivalent  positions: 

(b)  xyz;  xyz;  x-|-i  J-y,  z+|; 

i-x,  y-Hi,  z-f-i; 
x-Fiy-F^,  z;     ^-x,  |-y,  z;     x+f,  f-y,  z-Hi; 

f-x,  y-f-i  z-i-l; 
x+iy,  z-1-^;     l-x,  y,  z-l-l;     x-f-f,  f-y,  z-f-f; 

4~x,  y-rj,  Z-I-4; 
X.  y+i  z-l-l;     X,  |-y,  z+|;     x-f-^  f-y,  z-f-f ; 

i-x,  y+i  z+i 


56  THE   ORTHORHOMBIC  SPACE-GROUPS  cly-Y^. 

Space-Group  Cly, 

Two  equivalent  positions: 

(a)OOu;    i  i  u+i  (b)|Ou;    0,  i  u+|. 

Four  equivalent  positions: 

(c)  uOv;    uOv;    u+^,  ^,  v-f-^;    §-u,  ^,  v4-^. 

(d)  Ouv;    Otiv;    f,  u+^,  v+|;    I,  §-u,  v+^. 

Eight  equivalent  positions: 

(e)  xyz;  xyz;  xyz;  xyz; 
x+2,  y"r^)  z+2j    2~x,  2~y>  z-i-2>    x+2>  2~y>  Z+2> 

2-x,  y+i  z-hi 
Space-Group  C"- 

Fowr  equivalent  positions: 

(a)  OOu;    Uu;    0,  0,  u4-^;    i  i  u+|. 
(b)0|u;    §0u;    0,  i  u+l;    i  0,  u-h|. 

J5JzgfJ^f  equivalent  positions: 

(c)  xyz;  xyz;  x,  y,  z-}-|;        x,  y,  z-f-^; 

x+iy+2,  z+l;    ^-x,  ^-y,  z-Hi;    x+i  |-y,  z; 

i-x,y+^,  z. 
Space-Group  C|v. 

Four  equivalent  positions: 

(a)  OOu;    |0u;    i  i  u-f-^;  0,  i  u-}-|; 

(b)iuv;    |uv;    i  u-M,  v4-|;    i,  §-u,  v-|-|. 

^fgf^f  equivalent  positions: 

(c)  xyz;  xyz;  x+^,  y,  z;       |-x,  y,  z; 

x+iy+izH-^;    ^-x,  ^-y,  z+|;    x,  ^-y,  z+^; 

X,  y+l,  z-h^ 

B.  HEMIHEDRY, 
Space-Group  V^. 


One  equivalent  position : 

(a)  0  0  0.            (d)OOi.            (g)  OH- 

(b)iOO.            (e)HO.            (h)iH. 

(c)  0^0.            (f)   hOh. 

Two  equivalent  positions: 

(i)   uOO;    uOO.        (m)  OuO;    OuO. 

(q)  0  0  u;    OOu 

(j)  uO^;    uOi        (n)  Ou^;    Ou^ 

(r)  |0u;    |0u 

(k)  uH;    ti^O.        (o)  ^uO;    ^uO. 

(s)  OH;    0|u 

(1)  uH;    QH.        (P)|ui;    |ui 

(t)  Hu;    Hu 

Four  equivalent  positions: 

(u)  xyz;    xyz;    xyz;    xyz. 


THE    ORTHORHOMBIC   SPACE-GROUPS   V^-V^.  57 

Space-Group  V^. 

Two  equivalent  positions: 

(a)  uOO;    uOi  (c)  Oui;    0  u  f . 

(b)uH;    u*0.  (d)  inl;    i- u  f. 

Four  equivalent  positions: 

(e)  xyz;    xyz;    x,  y,  |-z;    x,  y,  z+i 

Space-Group  V. 

Two  equivalent  positions: 

(a)  OOu;    Hu-  (b)  0|u;    |Oti. 

Four  equivalent  positions: 

(c)  xyz;    x-fl,  |-y,  z;    |-x,  y-}-^,  z;    xyz. 

Space-Group  V*. 

Four  equivalent  positions: 

(a)  xyz;    x+|,  |-y,  z;    x,  y-|-i  |-z;    ^-x,  y,  z-|-|. 
Space-Group  V*. 

Four  equivalent  positions: 

(a)uOO;    uO^;    u+i  i  0;    ^-u,  i  i 

(b)  Oul;    Ouf;    |,  u+i  i;    |,  ^-u,  f. 

^tfif/if  equivalent  positions: 

(c)  xyz;  xyz;  x,  y,  |-z;  x,  y,  z-{-|; 
x+iy+l,  z;    x-l-il-y,  z;    ^-x,  y+i  ^-z; 

|-x,  l-y,  z+^. 

Space-Group  V. 

Two  equivalent  positions: 

(a)  0  0  0;    HO.  (c)  OH;    |oi 

(b)  ^00;    0^0.  (d)  HI;    OOi 

Four  equivalent  positions: 

(e)  uOO;    uOO;    u+i  i  0;    |-u,  i  0. 

(f)  uH;   QH;'  u+i  o,  |;   |-u,  o,  i 

(g)OuO;  OuO;  i  u+i  0;    i  i-u,  0. 

(h)  |u|;  |u*;  0,  u+i  |;    0,  |-u,  i 

(i)   OOu;  OOu;  Hu;    H  u. 

(j)   0|u;  Oiu;  ^Ou;    |0u. 

(k)  i^u;  44U;  44U;     44  u. 

Eight  equivalent  positions: 

(1)    xyz;  xyz;  xyz;  xyz; 

x+iy+iz;    x+il-y,  z;    |-x,  y+i  z;    ^-x,  |-y,  z. 


58 


THE   ORTHORHOMBIC   SPACB-GR0UP3   V^-V\ 


Space-Group  V^. 

Four  equivalent  positions: 


(a) 

000; 

HO; 

^01;    OH. 

(b) 

iOO; 

OiO; 

OOi;    Hi 

(c) 

ill. 

4    4    4; 

13    3. 

4   4   4> 

fH;   Hi 

(d) 

HI; 

Hi; 

IH;   IH. 

Eight  equivalent  positions: 

(e) 

uOO; 

uOO; 

u+i  i  0;    l-u,  i  0; 

uH; 

QH 

;    u+i  0,  1;    ^-u,  0,  -|. 

(f) 

OuO; 

OuO 

i  u+i  0;    i  l-u,  0; 

*u|; 

n^ 

;    0,  u+i  1;    0,  l-u,  i 

(g) 

OOu; 

OOu; 

i  0,  u+^;    -I,  0,  ^-u; 

Hu; 

Hu; 

0,  h  u+l;    0,  i  i-u. 

(h) 

Hu; 

ffu; 

3       3       l_i,.        1       1       l_,v 

4;     4>     2        '^J        4j     4)     2        "> 

ifu; 

fiQ; 

3     1     n-4-i-      i     ^     ii4-i 

4>     4)    U-t-2,        4>     4>     "^2' 

(i) 

|ui; 

3  ,,    3 

4  U    4, 

3       1_,,       3.        1       1_,,       1. 
4>     2        ">     4>        4>     2        ">     4> 

if,    3  . 
4  U  4, 

ftii 

;    i  u+i  \;    \,  u+i  f. 

Q) 

,,11. 
U   4    4, 

uff 

1_11       3       3.        1_,,       1       1. 
2        "j     4>     4>        2        ">     4;     4> 

f,  1   3 
U  4   4 

uH 

;    u+l.  f,  i;    u+i  i  i 

Sixteen  equivalent  pos 

itions : 

(k) 

xyz; 

xyz;                    xyz; 

xyz; 

x+i 

y+iz 

;    x+i^-y,  z;     ^-x,  y+i  z; 

^-x,  ^- 

-V,  7; 

x+i 

y,  z+l 

;    x+iy,  |-z;     ^-x,  y,  ^-z; 

^-x,  y, 

z+i 

X,  y+ 

iz+l 

;    X,  |-y,  l-z;     X,  y+i  |-z; 

X,  ^-y, 

z+i 

Space-Group  V*. 

Ttoo  equivalent  positions : 

(a)  000;    Hi 

(b)  iOO;    OH- 

Four  equivalent  positions : 


(c)  001;    HO. 

(d)  |0|;    0^0. 


(e)  uOO; 

(f)  uO|; 

(g)  OuO; 
(h)Ou|; 
(i)  OOu; 
(J)   0|u; 


uOO 
QO^ 
OuO 
Oui 
OOu 
0|u 


u+i  i  i;  l-u,  i  i 

u+i  i  0;  i-u,  i  0. 

i  u-l-i  i;  i  ^-u,  i 

i  u+i  0;  i  Ku,  0. 

2>    5>    U-|-2;  2>     2>    ^~U. 

i  0,  u-hl;  i  0,  i-u. 


£?tgi/if  equivalent  positions : 

(k)  xyz; 

x+i  y+i  z+l; 


xyz; 

x+i  i 


•y,  ^-z; 


xyz;  xyz; 

l-x,  y+i  |-z; 

^-x,  ^-y,  z-i-i 


59 


Space-Group  V*. 

Four  equivalent  positions: 

(a)  uOi;    l-u,  0,  f;    uH;    u+i  h  f. 

(b)  iuO;    f,  l-u,  0;    iQi;     f,  u+i  i 

(c)  Oiu;    0,  i  l-u;    IH;    h  I  u+|. 


jE^i'gf/if  equivalent  positions: 


(d)  xyz;  x,  y,  |-z;  |-x,  y,  z;         x,  |-y,  z; 

x+5,  y+iz+l;    x+i  |-y,  z;    x,  y+i  |-z;    |-x,  y,  z+|; 

C.  HOLOHEDRY. 

Space-Group  V^. 

One  equivalent  position : 


(a)  0  0  0. 

(d)Hi 

(k) 

OH. 

(b)  iOO.            (e)  0|0.            (h) 

hhh 

(c)  oo|.         (f)  HO. 

Two  equivalent  positions : 

(i)   uOO;    uOO.            (m)OuO; 

OuO. 

(q)  OOu; 

OOu 

(j)   uO|;    uOi            (n)  Oui; 

Oui 

(r)OH; 

OH 

(k)  u|0;    u*0.            (o)  HO; 

HO. 

(s)  |0u; 

|0u 

(I)  uH;   QH.         (p)Hi; 

huh 

(t)  1 1  u; 

IH 

Four  equivalent  positions: 

(u)  Ouv 

;    Ouv;    Ouv 

Ouv. 

(v)  H  V 

,    H  v;    Hv 

Hv. 

(w)  u  0  V 

uOv;    uOv 

uOv. 

(x)  u|v 

,    ufv;    uH 

uH* 

(y)  u  V  0 

uvO;    uvO 

uvO. 

(z)  uv| 

uv|;    uv| 

uvi 

Eight  equivalent  positions: 

(a)  xyz;    xyz;    xyz;    xyz;           xyz;    xyz; 

xyz;    xyz. 

Space-Group  V^. 

Two  equivalent  positions: 

(a)  000;    Hi                (c)  06| 

HO. 

(b)HO;    OH.              (d)HI 

0|0. 

Four  equivalent  positions: 

(e)  Hi 

IH;   IH;   Hi 

(f)   HI, 

fii;   HI;   IH. 

(g)  uOO 

tiOO;    l-u,  i  1;    u-Fi  i  i. 

(h)  uO| 

Q0|;    i-u,  §,  0;    u+i  I  0. 

(i)   OuO 

OuO;    i  i-u,  1;    i 

u+i  i 

(j)   Oui 

Ou|;    i  i-u,  0;    i 

u-hi  0. 

(k)  0  0  u 

OOti;    i  i  I-u;    i 

i  u+i 

(1)   OH. 

OH;  i  0, 

I-u;    i 

0,  u+i 

60 


THE   ORTHOREOMBIC   SPACE-GROUPS   yI-Y^ 


Space-Group  Yl  {continued). 
Eight  equivalent  positions: 


(m)xyz; 
^— X, 


xyz; 

■y,  l-z; 


xyz;        xyz; 

2-x,  y+i  z-\-\;    x-M,  l-y,  z-f-L' 
x+2>  y"i~2>  2~2;. 


Space-Group  V'. 

Two  equivalent  positions: 


(a)  OOi; 

(b)Hi; 
(c)  OH; 

(d)^Oi; 


oof. 

ill 

2   2   4- 

nil 

U  2   4* 
2  '-'  4- 


Four  equivalent  positions 


(i)  uOO 

tiOO; 

(j)  uH 

U  2  2> 

(k)  OuO 

OuO; 

(1)  ^u| 

h^h 

(m)  0  0  u 

OOu; 

(n)  H  u 

Hu; 

(o)  0  i  u 

OH; 

(p)  ^Ou 

^Ou; 

(q)  uv| 

;  uvf; 

(e)  0  0  0, 

OOi 

(f)  100 

|0i 

(g)  OH, 

0|0. 

C!  • 

(h)HI, 

HO. 

Lb. 

uOi 

uO|. 

uiO 

u^O. 

Ou| 

Ou|. 

^uO 

iuO. 

0,  0, 

l-u;  0, 

0,  u-Kl 

1  1 

2>  2> 

1  „ .  1 

^~U,   2, 

i  u+l 

0,  i 

l-u;  0, 

i  u-f-l 

i  0, 

l-u;  i 

0,  u+l 

uvf 

;  tivi 

^tg^^  equivalent  positions: 

(r)  xyz;  xyz; 

X  y,  |-z;    X,  y,  z+^; 

Space-Group  Yt. 


xyz; 


xyz; 


X,  y,  z-M;    X,  y,  \-z. 


Two  equivalent  positions : 


(a)  0  0  0;    ^^0 

(b)  10  0; 


2  2 

OiO. 


(c)  OH; 
(d)HI; 


|oi 
00  i 


Four  equivalent  positions: 


HO 


111 

4  I  5 


(e) 

(f) 

(g)  uOO 

(h)uH 
OuO 
|u| 


(i) 
(j) 


(k)  0  0  u 
(1)   0|u 


1 1  n. 

4  4  ^> 
13  1. 
I.l  f  > 

uOO; 

U  2    2, 

OuO; 

2^2, 

OOti; 
0|u; 


fiO; 


ffO. 

ff|. 
h  0; 
0,  h 
u,  0; 
0,  l-u,  I; 


fil; 

l-u, 

l-u, 
1    1 

2)     2 


u+l,  I,  0. 
u+l,  0,  |. 
I,  u+l,  0. 
0,  u+l,  i 


1  1    fS  .  ill, 

2  ^  U,        2    2"- 

lOu. 


|0u; 


Eight  equivalent  positions : 

(m)xyz;  xyz;  xyz;  xyz; 

l-x,  l-y,  z;    l-x,  y+l,  z;    x+|,  |-y,  z;    x+|,  y+|,  z. 


THE   ORTHORHOMBIC   SPACE-GROUPS   Y^-vl.  61 


Space-Group  Vh. 

Two  equivalent  positions: 

(a)  0  0  0;    OOi 

(d)HI; 

HO. 

(b)|00;    |0i 

(e)  Oui; 

Ouf. 

(c)  OH;    0|0. 

(f)   Iu|; 

1   ,-;    3 
^U  4. 

Four  equivalent  positions: 

(g)  uOO;    uOO;    uO|; 

uOi 

(h)  uH;   uH;   u-^O; 

u|0. 

(i)   Ouv;    Ouv;    0,  u, 

l-v;    0, 

ti,  v+l 

(j)   §uv;    |uv;    i  u, 

h-y;   h 

u,  v+i 

(k;)uv|;    uvf;    uv| 

;   uvf. 

jEzgf/i^  equivalent  positions: 

(1)    xyz;    xyz;    x,  y,  §-z;    x,  y,  z+|. 
xyz;     xyz;     x,  y,  z+i;     x,  y,  |-z. 

Space-Group  Vh. 

Four  equivalent  positions : 

(q\  iin-     iin-    111-     331 

yaj     44U,        44U,        442>        44^' 

/^Mlll.       131.       3in.       33n 
\"J    442>        442>        44'-'>        44'-'' 

(c)  uOO;    uO|;    |-u,  i  0;    u+i  i  i 

(d)  Oui;    Ouf;    i  i-u,  f;    |,  u+l,  i 

Etgf/i<  equivalent  positions: 

(e)  xyz;  xyz;  x,  y,  |-z;  x,  y,  z+|; 
l-x,  l-y,  z;    ^-x,  y+l,  z;    x+|,  ^-y,  z+i; 

x+5,  y+i  ^-z. 
A  slight  simplification  of  the  two  uniquely  defined  positions  [(a)  and  (b)] 

can  be  effected  if  the  origin  of  coordinates  is  changed  to  the  point  ( "o   "2^ )  ^^ 

this  first  set. 

Space-Group  V^. 

Two  equivalent  positions : 

(a)  iOO;    fOi  (c)  i  H;    f  i  0. 

(b)  foo;   ioi.  (d)  fH;   HO. 

Four  equivalent  positions: 

(e)  uOO;    u0|;    ^-u,  0,  0;    u+i  0,  ^ 

(f)  uH;    tii-O;    l-u,  I,  1;    u+i  §,  0. 

(g)  Oui;    Ouf;    Uf;    |ui 

(h)  iuv;    itiv;    f,  u,  |-v;    f,  u,  v+i 

^tgf/i<  equivalent  positions : 

(i)    xyz;  xyz;  x,  y,  |-z;  x,  y,  z+^; 

l-x,  y,  z;    §-x,  y,  z;    x+|,  y,  z+^;    x-f-^,  y,  l-z. 


62 


THE   ORTHORHOMBIC   SPACE-GROUPS   vl-vL 


Space-Group  Yl  (continued). 
By  shifting  the  origin  of  coordinates  to  the  point  ( "o  )  of  this  first  set,  these 


positions  become; 


Two  equivalent  positions: 

(a)  000;    AOi 

(b)  iOO;    OOi 

(c)  OH;    HO. 
(d)Hi;   oio. 

Four  equivalent  positions : 

(e)  uOO;    uOO;    ^-u, 

(f)  uH;   uH;   l-u, 

(g)  iui;    iuf;    fuf; 
(h)  Ouv;    Ouv;    |,  u, 

0,  1;    u+i  0,  h 
h  0.    u+i,  i  0. 

3  ,,   1 

4  U  4. 

l-v;    h  u,  v-h^ 

£'ig'/i<  equivalent  positions: 

(i)    xyz;    xyz;     ^-x,  y,  ^-z;     |-x,  y,  z-f|; 
xyz;     xyz;     x+|,  y,  z+§;     x+l,  y,  |-z. 


Space-Group  V^. 

Four  equivalent  positions: 


(a)  OiO; 

(b)  i  i  I; 

(c)  uOO; 
(d)Oui; 
(e)  |ui; 


OfO; 
i  3. 1- 

2   4   2; 

uO^; 
Ouf; 
iQf; 


0  i  1- 

^  4   2  > 

HO; 
ulO; 
0,  l-u,  f; 


"4   2- 

1  3  n 

2  4^' 

11  i  i 
U  2    2- 


1 

2> 


i  — 11      ^• 
2        ">     4> 


0,  u+i,  i 


Eight  equivalent  positions: 


(f)   xyz; 


xyz; 

X,  y+i  z; 


y,  i 


-z; 

z+l; 


X,  y,  z+^; 
X,  y+i  ^-z. 


X,  t-y,  z;    x,  yi-t,  z;    x,  f-y, 
The  unique  cases  can  be  simplified  by  transferring  the  origin  to  the  point  (  o^  )• 


Space-Group  Vu- 

Two  equivalent  positions: 
(a)  0  0  0;    HO. 
(b)00|;    Hi 

Fow  equivalent  positions: 


(c)  0|0; 
(d)OH; 


(e)  OOu 

(f)  Oiu 

(g)  uvO 
(h)uvf 


OOu; 

o§u; 

uvO; 
uv^; 


1  1  f, . 

2  2  "> 


?  f  u. 

^Ou. 


|0u; 

u+i  ^-v,  0; 

u+i 


i_ 


u,  v-fi  0. 


•V,  I; 


|-u,  v-f-l, 


Eight  equivalent  positions: 


(i)    xyz; 
xyz; 


x+i  ^-y,  z;     |-x,  y+i  z;    xyz; 


|-x,  y+i  z;    x+i  I 


i_ 


y.  z;     xyz. 


THE    ORTHORHOMBIC   SPACE-GROUPS   Vt-V".  63 

Space-Group  Vh". 

Four  equivalent  positions: 

/'o^lll•    313.    133.   Ill 
V<^y  444;    444}    444>    444- 

(V,\    11^-        111.   111.   Ill 
\")    444;   444;   444;   44  4* 

(c)  OOu;  Hu;  hh,   |-u;  0,  0,  u+i 
(d)0|u;  |0u;  i  0,  |-u;  0,  i  u+i 

^ipAi  equivalent  positions: 

(e)  xyz;  x+i  |-y,  z;  |-x,  y+i  z;  xyz; 

i-x,  |-y,  ^-z;  X,  y,  z+|;       x,  y,  z+l;      |-x,  |-y,  z+l. 

By  shifting  the  origin  to  f  ^'    o^'  o' )  ^^^  uniquely  placed  arrangements  can 

be  sUghtly  simplified. 

Space-Group  V". 

Four  equivalent  positions: 

(a)  OiO;    HO;    HO;    OfO. 
rMnii-    111-    131.    c\ii. 

\")    ^42;        242;        242;       "42' 

(c)  OOu;    Hu;    0|u;    |0u. 

(d)  iuv;    f  uv;    f,  |-u,  v;    \,  u+|,  v. 

Eight  equivalent  positions: 

(e)  xyz;  x+i  h-y,  z;    |-x,  y+i  z;    xyz; 

X,  h-y,  z;    i-X;  y,  z;  x+|,  y,  z;         x,  y+l,  z. 

The  unique  cases  can  be  simplified  by  placing  the  origin  at  the  point  (  2^  )• 

Space-Group  V". 

Two  equivalent  positions: 

(a)  OOi;    Hi  (c)  OH;    iof. 

(b)00|;    Hi.  (d)0H;    loi 

Four  equivalent  positions : 

(e)  g-C^«;    i-|u;    i  i  u-|-^;    0,  0,  Hu. 

(f)  6|u;    iOu;    i  0,  u+|;    0,  \,  \-m. 

(g)  uvi;    uvi;    u+i  Hv,  f;    |-u,  v+i  f. 

£'tgf/i<  equivalent  positions: 

(h)  xyz;  x+l,  |-y,  z;  Hx,  y+i  z;         xyz; 

X,  y;  ^-z;    l-x,  y+i  z-f-l;    x+i  HY;  z+^;    x,  y,  ^-z. 

The  unique  cases  can  be  simpUfied  by  changing  the  origin  to  (  2  )• 

Space-Group  Vh^ 

Two  equivalent  positions: 

(a)  OOu;    Hu-  (b)  0|u;    ^Oa. 


64  THE    ORTHORHOMBIC   SPACE-GROUPS   Vfa-Vh. 

Space-Group  Vh  (continued). 
Four  equivalent  positions: 


(c)  i  i  0; 

3  1  n- 

4   4  U,        4   4  U. 

(d)iH; 

3  11. 

4  4   2; 

13    1.        3    3    1 
4    4    2>        4    4    2- 

(e)  Ouv; 

Ouv; 

h,  l-u,  v;    h,  u+i  v. 

(f)   uOv; 

uOv; 

^-u,  1,  v;    u+i  1,  v. 

Eight  equivalent  positions : 

(g)  xyz;     x+i  |-y,   z;     ^-x,  y+^,  z;     xyz; 
l-x,  |-y,  z;    xyz;    xyz;    x+i  y+i  z. 

The  unique  cases  can  be  simplified  by  changing  the  origin  to  (  ^'  o^  j. 
Space-Group  V^h  • 

Four  equivalent  positions: 

(rt\  ini-    ^ii-    113.    ani 

V.ct;     4'-'4>        424;        424>        4'-'4' 

Ch'i^n^'    111-    311.    10^ 

\'->J     4'-'4>        424;        424;        4^4' 

(c)  OOu;    Hii;    i  0,  |-u;    0,  i  u-|-i 
Eight  equivalent  positions : 

(d)  xyz;  x+i  §-y,  z;    ^-x,  y+f,  z;    xyz; 

l-x,  y,  §-z;    X,  y+i  z+l;      x,  |-y,  z+l;    x+i  y,   |-z. 

By  changing  the  origin  to  the  point  f  -^'  ^  j  the  unique  cases  are  simplified. 

Space-Group  V^^ 

Four  equivalent  positions: 

(a)  000;    HO;    OH;    §0i 

(b)  HI;    0  0  i;    i  0  0;    010. 

Eight  equivalent  positions : 

(c)  xyz;    x+i  |-y,  z;    x,  y-{-^,  |-z;    |-x,  y,  z+i 
xyz;    l-x,  y+l,  z;    x,  |-y,  z-|-§;    x-|-i  y,  |-z. 

Space-Group  Y'^. 

Four  equivalent  positions : 

(a)    Xlf).      3  in.      111.      Ill 
\aj     44  W,        44  U,        442;        44   2- 

CM    ill-      114'      3.  3  ().      lln 
W>'    442;        44l;        44*^;        44'-'' 

(c)  Ouv;    h  l-u,  v;    0,  u+i  |-v;    i  u,  v-}-^ 
^z'gfAf  equivalent  positions: 

(d)  xyz;    x+i  |-y,  z;    x,  y+|,  |-z;    ^-x,  y,  z+^; 
l-x,  l-y,  z;    xyz;    x+|,  y,  z+|;    x,  y+|,  ^-z. 


The  unique  cases  are  simplified  when  the  origin  is  changed  to    (  ^ 


(f  ?)• 


THE    OETKOREOMBIC   SPACE-GROUPS 


V^^V^,«. 


65 


Space-Group  Vh^. 

Four  equivalent  positions : 

(a)  0  0  0;    0  0 1;    HO;    Hi 

(b)  |00;    iOi;    0|0;    OH- 

(c)  Ou|;    Ouf;    I  u+i  h 

Eight  equivalent  positions: 


1  i_„ 

2>  2   "> 


ill. 

4  4  2; 


/'rl^lln-  3.  If).      311. 

3  3  n.  3  1  A. 

4  4  U,  4  4  "> 

(e)  uOO;  uOO; 

uO^;  uO|; 


3  3  1. 

4  4  2> 

111 
4  4  2- 


1   ,,   1   1 
2~U,  7j,  5 


"T^2>  2>  1 

u+i  i  0;  l-u,  i,  0 


(f)  Ouv;  0,  u,  ^-v;  i  u+i  v;  i  u+i 


Ouv;  0,  u,  v-hh;    h,   i 


■u,  v;  f, 


^-u, 


i-v; 

v+i 


(g)  uvi;  uvf;  u+i  v+^,  i;  u+f,  |-v,  f; 
uvi;  i-u,  l-v,  f;  |-u,  v+i  |. 


uvf; 


Sixteen  equivalent  positions : 

(h)  xyz;  xyz;  x,  y,  ^-z;  x,  y,  z+|; 
xyz;  xyz;  x,  y,  z+|;  x,  y,  |-z; 
x+l,  y+i,  z;    x+i  |-y,  z;    ^-x,  y+i  §-z; 

^-x,  i-y,  z+^; 
2-x,  i-y,  z;    ^-x,  y+l,  z;    x+i  §-y,  z+|; 

x+i,  y+i  l-z. 
Space-Group  Vh^. 

jPowr  equivalent  positions : 


(a)  iOO;    10^;     HO; 

(b)  fOO;    10  h    IhO; 

Eight  equivalent  positions : 


111 

4    2    2- 

3  11 

4  2    2- 


(c)  OH 
HO 

(d)  u  0  0 


uH 


(e)  Oui 
Ouf 

(f) 


i  u  V 
iu  V 


OfO;  OH;  Ofl; 

iin-  lai.  Ill 

24  ">  242>  24    2' 

uOi; 


u-iO; 

Uf; 


Hu,  0,  0;    u+i  0,  ^; 
i— 11   i   A-    ii4-i   1   n 

i  l-u,  f;    0,  uH,  i; 

2,  u+l,  i;    0,  l-u,  f. 
i  u,  i-v;    f,  u+i  v;    f,  u+i  |-v; 
i  %  v-|-§;    I,  §-u,  v;    i|-u,  v+i 


|u 


Sixteen  equivalent  positions: 


xyz; 


(g)  xyz; 

x+i  y+l,  z;    x+l,  Hy,  ^ 


X,  y,  f-z; 


X,  y,  z+i; 


X,  y,  z;     ^-x,  y,  z;     x+i  y,  z+^;     x+i  y,  ^- 


z: 


l-x,  y+i  Hz; 

^-x,  Hy,  z+l; 


X,  Hy,  z;    X,  y+l,  z;    X,  Hy,  z-H;    x,  y-f-i  f 


•z. 


66 


THE   ORTHORKOMBIC   SPACE-GROUPS   V 


Space-Group  V". 

Two  equivalent  positions: 


(a)  0  0  0; 

HO. 

(c)  OH 

|0|. 

(b)  10  0; 

0|0. 

(d)IH 

00|. 

Four  equivalent  positions : 

(e)  HO; 

HO; 

3  1  n.      3  3  n 

4  4  U,        4   4  U. 

(f)   HI, 

13    1. 

4    4    2) 

3  11.        3    3    1 

4  4  5>        4   4   2- 

(g)  uOO 

uOO, 

u+l,  1,  0;    1- 

-u,  1,  0. 

(h)uH 

fill. 

U  2    2  J 

u+l,  0,  1;    1- 

-u,  0,  |. 

(i)    OuO 

OuO; 

h  u+l,  0;    1, 

l-u,  0. 

(i)  huh, 

1  fi  1 . 

2^2, 

0,  u+l,  1;    0, 

l-u,  |. 

(k)  OOu, 

OOu; 

2  1  U;       2  2  u. 

(1)   0|u, 

0|u; 

|0u;    |0u. 

Eight  equivalent  positions: 

(m)Hu 

Hu, 

Hu;    Hu; 

IfQ 

fiu, 

13,,.        1    1  fi 
4  4  U,       4   4  U. 

(n)  Ouv 

;    Ouv 

;    1,  u+l,  v; 

1,  u+l, 

v; 

Ouv 

;    Ouv 

;    1,  l-u,  v; 

1,  l-u, 

V. 

(o)  uOv 

;    uOv 

;    u-f  1,  1,  v; 

l_n     1 

2        "j     2> 

v; 

uOv 

;    uOv 

;    u+l,  1,  v; 

1          n        1 

2— U,    2> 

V. 

(p)  u  v  0 

;    uvO 

;    u+l,  v+l,  0; 

l-u,  1- 

-V, 

0; 

uvO 

;    uvO 

;    u+l,  l-v,  0; 

l-u,  V- 

fl, 

0. 

(q)  u  V  1 

;    u  V  ^ 

;    u+i  v+l,  1; 

1         n       1 
2— U,     2" 

-V, 

1; 

uv| 

;    u v| 

u+l,  l-v,  1; 

l-u,  V- 

fl, 

1 

2' 

Sixteen  equiva 

ilent  pos 

itions 

(r)  xyz; 

xyz; 

xyz;        xyz; 

xyz; 

xyz; 

xyz;         xyz 

x+l, 

y+l,  z 

;    x+i  l-y,  z; 

l-x,  y+ 

1,  z 

;  l-x, 

l-y,  z; 

^  — X, 

l-y,  z 

;    l-x,  y+l,  z; 

v-4-i    i  — 

A-r2,  2 

y,  z 

1       X  +  2, 

y+l,  z. 

Spa  ;e-Group  V1°. 

Four  equivale 

Qt  positi 

ons: 

(a)  0  00 

00^; 

HO;    HI. 

(b)|00 

|0|; 

0^0;    OH. 

(c)  OOi 

OOf; 

IH;   Hi 

(d)OH 

OH; 

|0i;    |0i 

(e)  iH 

HI; 

fH;   Hi 

(0    Hf, 

111. 

4   4   4; 

111.   Ill 

4   4   4  >        4   4   4* 

£Jifif/i<  equivalent  positions: 


(g)  uOO 
uOO 

(h)  0  u  0 
OuO 


uO|;    u+l,  I,  0 


UU  2, 

Oui; 
Ou|; 


l-u,  I,  0 
I,  u+l,  0 
I,  l-u,  0 


1    1 . 

2,    ?> 
1       1 


U  +  l, 

l-u,  I, 
I,  u+l,  I; 
I,  l-u,  |. 


THE    ORTHORHOMBIC   SPACE-GROUPS   Vh^-v". 


67 


Space-Group  Y^^  {continued). 


(i)   OOu;  Hu 

OOti;  Hu 

(j)   0|u;  lOu 

O^u;  —- 

(k)  iiu; 

3    3,,. 

(1)    uvi;  uv 


2  u  u 

1    3  ,-, 
4   4  U 

3  1   ,-, 

4  4    U 

i-r   3 


uv|; 


ti  vf 


0,  0,  u+l; 

0,  0,  i-u; 

0,  i  u+l; 

VJ,  2j  2         '^J 

1  1  l—u- 

4»  4>  2         "^J 

3  3  1_,,. 

4>  4)  2        "^J 

1  —  11  l_v     !• 

2  U)     2        V,     4, 


Sixteen  equivalent  positions : 
(m)  xyz;  xyz; 


2>  2>  2   "^' 

h   0,  u+l; 
i  0,  i-u. 

1   3   1,4.1. 
4)  4>  "T^2> 

3   1   ii_L.l 
4)  4>  U-t-2. 

u+l,  i-v,  f; 
l-u,  v+i  f. 


xyz; 


xyz; 


X,  y,  t-z; 


X,  y,  z+l; 


X,  y,  z+^;    X,  y,  |-z; 


x+i  y+i,  z;    x+i  ^-y,  z;    ^-x,  y+|,  z; 

i-x,  |-y,  z; 
2~x,  2~yj  2~z;  2— X,  y+^'j  z+2 j  x+j,  2~y>  z-|-2> 

x+i,  y+i  i-z. 


Space-Group  Vh^ 

jPowr  equivalent  positions: 

(a)  0  0  0; 

JOO; 

HO;    o\o. 

(b)OOi; 

5  0  1; 

111.      nil 

2    2    2»       U  2    2- 

(c)  iOO; 

fOO; 

fH;   HO. 

(d)HI; 

3  11. 

4  2    2; 

101;    ioi 

(e)  OiO; 

OfO; 

HO;    HO. 

(f )      2   4   2  5 

13    1. 

2    4    2» 

n  a  1-    nil 

^  4   2»       ^4   2- 

(g)  i-iu, 

13,-.. 
4   4  U, 

Hu;    Hu. 

Eigr/i<  equivalent  posit 

ions: 

(h)  u  0  0 

ulO 

Hu,  h  0;    u+i  0, 

0; 

tiOO 

uiO 

u+l,  i  0;    Hu,  0, 

0. 

(i)    uH 

uOi 

Hu,  0,  \;    u+l,  1, 

h 

aH 

uOi 

u+2>  0,  1;    Hu,  ^, 

1 

2- 

(j)   OuO 

HO 

i  Hu,  0;    0,  u+i 

0; 

OuO 

HO 

i  u+i  0;    0,  Hu, 

0. 

(k)  hnh 

Ou^ 

0,  Hu,  \',    i  u+i 

1 . 

\^\ 

Ou^ 

0,  u-hi  i;    1,  Hu, 

1 

2' 

(1)    OOu 

2-Ou 

OH;   Hu; 

OOu 

Hu 

OH;   H u. 

(m)  J  u  V 

f  U  V 

;    i  u-l-i  v;    i  u+i 

v; 

J  U  V 

1  U  V 

;    i  Hu,  v;    \,  Hu, 

v. 

(n)  u  i  V 

uf  V 

;    u-l-i  f,  v;    u-l-i  \, 

v; 

ti  J  V 

uf  V 

;    Hu,  i  v;    Hu,  i 

V. 

Sixteen  equiva 

lent  po£ 

jitions: 

(o)  xyz; 

xyz;                   xyz; 

xyz; 

i-x, 

y,  z; 

Hx,  y,  z;        x+i 

y,  z; 

x+i  y,  z; 

x+i 

y+i  z 

;  x+iHy,  z;  Hx, 

y+i  z; 

Hx,  Hy,  z; 

x,|- 

■y,  z; 

X,  y+l,  z;       x,  H 

y,  z; 

X,  y+i  z. 

68 


THE   ORTHORHOMBIC   SPACE-GROUPS   Vh^-Vh^ 


Space-Group  Y^. 

Four  equivalent  positions : 


(a)  0  0  0;    i 

(b)  iOO; 


-    0    i- 

2   ^  2  > 


1  1  0- 

2  2   "> 

OOi;     0|0; 


"2    2- 

111 
2    2^' 


Eight  equivalent  positions: 


(c)  |0i 

3  11 

4  2    4 

(d)  0  H 
(e) 


13    1 

2    4    4 


(f) 


uOO 
uOO 
OuO 
OuO 
(g)  OOu 

OOti 

1 


(h) 


_  u 

3  3  „ 

4  4" 


0  1-        3  n  3. 
^  4  J        4^4) 


1  3. 

2  4  > 

3  3 

4  4  J 
1  3  . 
4  4> 


Oil;     Oil:     0 


1  3. 

2  4> 

1  3 

4  4> 

3  3. 

4  4; 


u+i  0,  i; 

2~U,     0,     2) 

0,  u+i  i; 


'-'  4> 

1  1 

2  4- 

3  1  . 

4  4  > 
1    1 
4   4- 

1 

2> 
1 
2? 


0, 
0, 
0, 


-u,  i; 
u+l; 


1    3 


t-u; 

-    -   IT         13 

3  1  ri  •        3       1 

4  4   U,        4,     4, 


i  u+l,  0; 
2>  2~u,  0; 

I,  0,  u+l; 

2^>     0,     2        ^5 


l-u; 


1_„.        3       3 
2        U,        4,     4, 


1,   1    1. 

U  2    2  > 

fl    i    i 

U  2    2* 

1  „    1. 

2  U  2  ) 

i  f]   i 
2    U  2- 

1  1   „. 

2  2    "^^ 

Hu. 
u+l; 
u+i 


Sixteen  equivalent  positions : 


(i)   xyz; 


xyz; 


xyz; 


xyz; 


l-x,  y,  |-z;  l-x,  y,  z+l;  x+i  y,  z+|;  x+i,  y,  |-z; 
x+iy+l,  z;  x+l,  |-y,  z;  |-x,  y+|,  z;  f-x,  |-y,  z; 
X,  l-y,  |-z;    X,  y+i  z+^;    x,  §-y,  z+|;    x,  y+i  ^-z. 


Space-Group  Vh^ 

/^owr  equivalent  positions: 


(a)  00  0 

1  1  n. 

2  2  ^j 

1  n  1 

2^2) 

OH. 

(b)^OO 

0|0; 

OOi, 

Hi 

t  equivalent  positions: 

(c)  OH 

0  13. 
^  4   4j 

n  1 3 

U  4   4 

Ofi; 

Hi 

113. 
2    4    4) 

13    3 

2    4    4 

Hi. 

(d)  \oi 

1  n  a- 

4  '-'  4» 

fOf 

fOi; 

3  11 

4  2    4 

3  13. 

4  2    4) 

iH 

iH. 

(e)  HO 

ifO; 

fiO; 

HO; 

111 

4   4   2 

H^; 

ii^; 

iH. 

(f)  Hi 

13    3. 

4    4   4) 

3  13. 

4  4    4) 

Hi; 

3  3   1 

4  4   4 

fii; 

ifi. 

111 
444. 

(g)  uOO 

u+i 

0,  ^; 

u+i  i 

0; 

u 

uOO 

l-u, 

0,  1; 

l-u,  h, 

0; 

u 

(h)  0  u  0 

0,  u+i  h; 

h,  u+i 

0; 

^ 

OtiO 

0,  h- 

-u,  h; 

i  Hu, 

0; 

^ 

(i)   OOu 

0,  i 

n+h 

i  0,  u+^; 

1 

2 

OOu 

0)  i 

h-n; 

i  0,  H 

-u; 

^ 

u 


^; 


1 
2. 
i. 

2) 

u|. 
^u; 
^u. 


TEE    CRTEORKOMBIC   SPACE-GROUPS   Vh^-vl*. 


69 


Space-Group  V^^  (continued). 
Sixteen  equivalent  positions: 
(J) 


(k) 


(1) 


(m) 


(n) 


(o) 


i  i  11 

4   4    U 

13,,. 
4   4  U, 

i  h  u+l; 

1 

4> 

3 

4j 

u+L- 

IfQ 

3  1  n  • 

4  4   U, 

3       3       1 

4>     4>     2~ 

-u; 

3 

4) 

1 

4) 

1 

2- 

-u; 

flu 

3  1,,. 

4  4  U, 

3       3      ,i_J_l. 
4>     4)     Ui-2, 

i 

1 

47 

u+l; 

HQ; 

13.-,. 
4    4    U, 

111 

4>     4>     2 

-u; 

1 
4> 

3 

47 

^- 

■u. 

1    n    1 
4   U   4, 

1  n  1 . 
4  U  4, 

i  u+i 

1 . 

4  > 

1 
47 

u- 

f|7 

f; 

ftif, 

3  ,-,    1  • 

4  U   4, 

i  l-u, 

3. 
4> 

3 

4) 

1 

2  ■ 

-u, 

1 . 

47 

fuf, 

3  ,,    1  . 

4  U  4  , 

3      ,,_Ll 
4,    U-1-2, 

3  . 

4> 

3 

4> 

u- 

H, 

1 . 

47 

1   Ti    1 

4  U   4, 

iQf; 

1       1_,] 

4>     2        "> 

1  . 

4  7 

1 
47 

1 

2  " 

-u, 

3 

4- 

11    1    1 
U  4    4, 

uH; 

u+i  h 

1. 

4» 

u 

■f^ 

1 

7      47 

3. 

47 

f,    3    3 
U  4    4, 

f,    3    1  . 
U  4   4, 

1  —  11    ^ 

2        ")     4> 

3  . 

47 

1 

2 

-u 

3 

7      47 

1  . 
4  7 

,,3    3 
U  4   4, 

uH; 

u+i  I, 

3  . 
4) 

u 

H 

3 

7      47 

1  . 

4> 

vi   1    1 
U  4   4) 

GH; 

l-u,  i, 

1  . 
4> 

1 

2 

-u 

1 

7      47 

3 
4- 

Ouv 

0,  u+i,  v+^; 

1 
2> 

u+i 

v; 

1 
27 

u, 

v+§; 

Ouv 

;   0,  1- 

-u,  l-v; 

1 
2> 

1 

2 

u, 

v; 

1 

27 

U7 

l-v; 

Ouv 

0,  u-Hi  ^-v; 

i 

u+i 

v; 

1 

2> 

u, 

i-v; 

Ouv 

0,  1- 

■u,  v+l; 

1 
2j 

1 

2~ 

U7 

v; 

h 

U7 

v-l-i 

uOv 

;   u+i 

0,  v+l; 

u 

4-i 

i 

v; 

U7 

1 

27 

v+^; 

uOv 

u+i 

0,  l-v; 

u 

H, 

1 

27 

v; 

U7 

h 

l-v; 

uOv 

;   l-u, 

0,  l-v; 

^ 

-u, 

h 

v; 

U7 

1 

27 

l-v; 

uOv 

;   l-u, 

0,  v+^; 

1 

2 

-u, 

h 

v; 

u, 

1, 

v-h|. 

uvO 

;   u+i 

v-}-i  0; 

u 

4-i 

V, 

h; 

u, 

v+i  1; 

uvO 

;   u+i 

i-v,  0; 

u 

■fi 

V, 

1. 

2> 

u, 

1. 
2 

-V,  1; 

uvO 

;   ^-u, 

v+i  0; 

1 

1- 

-u, 

V, 

h 

U7 

V 

H,  1; 

u  vO 

;   l-u, 

l-v,  0; 

f 

-u, 

V, 

1 . 

27 

Q, 

1 

2" 

-V,  |. 

Thirty-two  equivalent  positions: 

(p)  xyz;   xyz;  xyz;   xyz; 

xyz;   xyz;  xyz;   xyz; 

X+I7  y+i  z;  x-}-^,  |-y,  z; 

l-x,  ^-y,  z;  l-x,  y+l,  z; 

x+l,  y,  z+l;  x-l-l,  y,  |-z; 

l-x,  y,  |-z;  l-x,  y,  z-\-i; 

X,  y+l,  z+l;  X,  l-y,  l-z; 

X,  |-y,  |-z;  X,  y-fi  z+l; 


l-x,  y+i,  z; 
x+27  2~y7  z; 
2— X,  y,  ^  — z; 


f-X7  l-y,  z; 

x+l,  y+i  z; 

l-x,  y,  z-f-§; 
X+I7  y,  z+l;  x+i  y,  §-z; 
X,  y+i  l-z;  X, i-y,  z+i* 
x>  l-y,'  z+l;  X,  y+i  i-z. 


Space-Group  V  h*. 

Eight  equivalent  positions: 

(a)  0  0  0;    HO;    10^;    0|i 


ill. 
4447 


13   1. 
4    4    47 


111. 

4    4    4  7 


(b)  §0  0;    0^0;    OOi;    ^ 


1  1  !• 
4  4  4; 


13   1, 

4   4  2; 


111. 
4   4  4; 


2   2  ; 

113 

444. 

111. 

2   2   2; 

1  3  3. 

444. 


70 


24     ,.25 


THE   ORTEOEEOMBIC   SPACE-GROUPS   Vh-Vh. 


Space-Group  Vh*  (continued). 
Sixteen  equivalent  positions: 


111 

S  5  8 

13  3 

8  8  8 

13  7 

8  8  8 
5  15 

8  8  8 

5  5  5 

8  8  8 

5  7  7 

8  8  « 

17  3 

8  ?  8 
111 

8  8  8 


111- 
8  8  8  7 
111. 
8  8  8> 
3  5  7. 
8  8  8) 
111. 
8  8  8  > 
5  1  3. 
8  8  8  > 

7  5  1. 

8  8  8  > 
113. 
8  8  8> 
1  5  1. 
8  8  8) 


7  17. 
¥88) 

111. 

8  8  8) 
15  5. 
8  8  8) 
3  15. 
8  8  8) 
3  5  3. 
8  8  8) 
115. 
8  8  8) 
111. 
8  8  8) 
7  3  1. 
¥  8  8) 


111. 
8  8  ¥) 
5  5  1. 
8  8  8) 
1  5  1. 
8  8  8) 
13  5 
8  8  8* 
3  3  5. 
8  8  8) 

114. 

8  8  8) 

111. 
8  8  8) 

111 

8  8  8- 


(c) 


(d) 


(e) 


(f) 


(g) 


Thirty-two  equivalent  positions: 

(h)  xyz;        xyz;        xyz;        xyz; 

i-x,  i-y,  i-z;    i-x,  y-\-h  z+i;    x-f-i,  l-y,  z+i; 

x+i  y+i  i-z; 
x+l,  y+l,  z;    x-\-i,l-y,z;    |-x,  y-f-^,  z;    ^-x,  ^-y,  z; 
l-x,  f-y,  i-z;    l-x,  y-M,  z+i;    x+f,  f-y,  z-f-i; 

x+f,  y+f,  i-z; 
x+l)  y)  z-f-l;    x-l-iy,  ^-z;    i-x,  y,  ^-z;    J-x,  y,  z+|; 
f-x,  i-y,  f-z;     f-x,  y+i,  z-|-f;    x-|-f,  i-y,  z-h|; 

x+l,  y+i  f-z; 
X,  y+5,  z-}-^;    X,  |-y,  |-z;    x,  y+i  ^-z;    x,  |-y,  z-f-|; 
i-x,  f-y,  f-z;    J-x,  y-l-f,  z-f-f;    x-hi  f-y,  z+f; 

x-ff,  y+f,  f-z. 


uOO 

u- 

■f-l,  0,  1; 

u+i  h  i; 

u+i  i,  f; 

uOO 

1. 

2 

-u,  0,  1; 

1 

4" 

-11      i      i- 

")     4)     4) 

3 

4  " 

-U)  i  f; 

uH 

uH)  i  0; 

u+i  i  1; 

ii-J_l      1       1. 
•^    r4)     4)     4) 

uH 

^ 

-u,  i  0; 

i_ 

4 

-u,  i  1; 

1_ 

4 

-u,  i  h 

OuO 

0, 

u+i  i; 

1 
4) 

u+i)  f; 

1 
4) 

u+l,  f; 

OuO 

0, 

l-u,  1; 

i) 

i-u,  i; 

f) 

f-u,  f; 

hul 

i 

u+i  0; 

1 

4) 

u+i,  f; 

1 

4) 

u+f,  i; 

HI 

1 

2) 

l-u,  0; 

1 

4) 

i-u,  f; 

3 
4) 

l-u,  h 

OOu 

0, 

h  u+l; 

1 

4) 

i  u+i; 

1 

4) 

I  u+f; 

OOu 

0, 

i  l-u; 

1) 

i  i-u; 

i) 

f)  f-u; 

Hu 

i 

0,  u+l; 

f, 

1  iij-i- 

4)    Ui-4, 

f) 

1,  u+f; 

Hu 

,  i 

0,  l-u; 

3 
4) 

3       1        ,,. 
5)     4— U, 

f, 

f,  f-u. 

Space-Group  V^^. 

Two  equivalent  positions: 

(a)  0  0  0; 

Hi 

(c)  0  01; 

HO. 

(b)HO; 

OH. 

(d)| 

01 

0|0. 

Four  equivalent  positions: 

(e)  uOO; 

uOO; 

u- 

H, 

hh 

1- 

-u,  i  i 

(f)   uO|; 

tiOl; 

U  +  l) 

h  0; 

1- 

-u,  1,  0. 

(g)  OuO; 

OuO; 

i 

u+l)  1; 

1 

2) 

l-u,  |. 

(h)Ou|; 

onh; 

i 

u+i  0; 

1) 

l-u,  0. 

(i)   OOu; 

OOVl; 

i 

h 

u+l; 

i 

1,  l-u. 

a)  OJu; 

OH; 

i 

0, 

u+J; 

1, 

0,  l-u. 

THE    ORTHORHOMBIC   SPACE-GROUPS 


Y\'-y'l 


71 


Space-Group  Vh^  (continued). 
Eight  equivalent  positions: 


(k)Hi; 

3  3  3, 

4  4  4> 

13  3. 
4  4  4> 

3  11. 

4  4  4; 

(1)  Ouv 

;  Ouv; 

Ou  V 

;  Ouv; 

(m)  u  0  V 

;  uOv; 

uOv 

,    uOv; 

(n)  u  V  0 

;  uvO; 

uvO 

;  uvO; 

3  13.    331. 

4  4  47    4  4  4) 

13  1.    113 

4  4  4j   4  4  4- 

i  u+i  v+l; 

i  u+l,  i-y; 

h,   Hu,  l-v; 

h  l-u,  v-H. 

u+l,  i  v+l; 

u+i  i  i-v; 

Hu,  h,  l-v; 

Hu,  i  v+|. 

u+i  v+^,  1; 

ii_l_l    1    lr    !• 

Ui"2>  2~V,  5, 

1    11   1    -jr   1  • 
2~U,  2    V,  2; 

f-u,  v+i  i 

Sixteen  equivalent  positions : 

(o)  xyz;        xyz;        xyz;        xyz; 
xyz;        xyz;        xyz;        xyz; 

x+l,  y+i  z-H;    x+i  |-y,  |-z;    |-x,  y+i  |-z; 

2~x,  2~y>  z+fj 
2~x,  2~y>  2~z;    2'~x,  y+2)  z-f-a;    x+2,  2~y>  z+g-; 

x+i  y+i  l-z. 
Space-Group  V^". 

T^'owr  equivalent  positions : 


(a)  0  0  0; 

0  0^; 

HI; 

HO 

(b)  iOO; 

1  n  i- 

2  "  2> 

OH; 

0|0 

(c)  OOi; 

oof; 

HI; 

Hi 

(d)OH; 

OH; 

lOf; 

|oi 

£^?gf/i<  equivalent  positions: 


(e)  HO, 

1  3  A. 
4  4  *J) 

3  3  1. 

111. 

4  4  2) 

4  4  2) 

(f)  uOO 

uOA; 

uOO 

uO|; 

(g)  OuO 

Ou|; 

OuO 

Ou|; 

(h)  0  0  u 

Hu; 

OOu 

HQ; 

(i)  0|u 

|0u; 

0|u 

|0u; 

(J)  uvi 

;  uvf; 

Gvi 

;  uvf; 

3  1  n-     3  3  n- 

4  4  U,        J   4  U, 

13    1.        Ill 
7  4    2)        442' 

u+i  h  0; 
i_i,   1  n- 

2        '^)     2)    ") 

i  u-fi  0; 
i  Hu,  0; 
i  i  u-H; 

1         1  1_1T 

2)    ^>    5        ") 

I,  0,  u+l; 
i  0,  Hu; 
u+l,  Hv,  I; 
Hu,  v+l,  i; 


11-4.1    1    1. 

U^2)     2)     2) 

i— 11    i    i 

2        ")     2)     2- 

1      „_Ll       1. 
1)    U-f-2)    ?) 

i  Hu,  i 
0,  0,  u+i; 
0,  0,  Hu. 
0,  i  u+l; 
0,  i  Hu. 
u+i  v-l-i  f ; 
Hu,  Hv,  f. 


Sixteen  equivalent  positions : 
(k)  xyz;  xyz; 


xyz; 


xyz; 


X,  y,  Hz;    x,  y,  z+l;    x,  y,  z+|; 


x+i  y+i  z-\-^;    x-i-i  Hy,  Hz; 


Hx,  Hy, 


z; 


y,  t-z; 

2~x,  y-r^,  2~z; 

2~x,  ^— y,  z-|-f; 
Hx,  y+l,  z;    x+i  Hy)  z;    x-fi  y-f|,  z. 


72 


Space-Group  \^h  • 

Eight  equivalent  positions: 

(a)  000 

•    001; 

|00;    0^0; 

111 

2    2    2 

HO; 

OH;   ioh 

(b)  1  H 

Hi; 

113.        3    11. 
4   4   4>       4  4   4; 

3    3    3 

3    13. 

3    3    1.       13    3 

4  4   4) 

4   4    4> 

4   4   4>        4    4   4- 

(c)  uOi 

;   uH; 

u+i  0,  i;    u+i  i  f; 

iiOf 

;   QH; 

Hu,  0,  f;    Hu,  h  h 

(d)iuO 

Uh 

h  u+i  0;    1,  u+i  i; 

f  uO 

luh; 

i  Hu,  0;    i  Hu,  i 

(e)  0|u 

Hu; 

0,  h  u+i;    i  f,  u+i; 

Ofu 

Hu; 

0,  i  Hu;    i  i  Hu. 

Sixteen  equivalent  positions: 

(^)   xyz, 

X,  y,  i- 

-z;    Hx,  y,  z;    x,  Hy,  z; 

xyz;    X,  y,  z+|;    x+i  y,  z;    x,  y+i  z; 
x+l,  y+i  z+l;    x+i,  Hy,  z;    x,  y+i  Hz; 

2~x,  y,  z+5; 

Hx,  Hy,  Hz;    Hx,  y+l,  z;   x,  Hy,  z+i; 

x+i  y,  Hz. 


Space-Group  Vf . 

Fowr  equivalent  positions: 


(a)  OOi; 

HI; 

OH; 

HI 

(b)Hi; 

OOf; 

HI; 

OH 

(c)  HO; 

HI; 

HO; 

IH 

(d)HO; 

111. 

4   4   2, 

HO; 

fH 

(e)  0|u; 

^,     4, 

Hu; 

1       3       , 
2,     4,    ^ 

Ezfirj'if  equivalent  positions: 


(f)  uO| 
uOi 

(g)  Iu0 
f  uO 

(h)  Ouv 

|u  v 

(i)  ujv 


11  i  i- 

"2  4, 


u+i  0,  f;  u-hl  I  f; 
uH;  Hu,  0,  I;  Hu,  i  f. 

iul;  i,  Hu,  0;  J,  u-H,  |; 
f  ul;  f,  Hu,  0;  i  u-f-i  i 
0,  u,  Hv;  i  Q,  v-fl;  h  u-H,  v-H; 
0,  Hu,  v;  I,  Hu,  ^;  0,  u-hi  Hv. 
u,  i  Hv;  u-H,  h  v;  u-l-i  f,  v-f-^; 
v; 


fi  4  i  — 

u,  J,   2 


Hu,  i  v;  Hu,  f,  v-F^. 


Sixteen  equivalent  positions: 


(j)  xyz;    X,  y,  Hz;  Hx,  y,  z;    x,  Hy,  z; 

Hx,  Hy,  z;    xyz;    X,  Hy,  z;    x+i  y,  z; 
x+i  y+iz-H;    x+iHy,  z;    x,  y+i  Hz;    Hx,  y,  z+|; 
X,  y,  Hz;    Hx,  y+i  z+^;        x-|-|,  y,  z+|;    x,  y-f-l,  Hz. 
The  unique  cases  can  be  simplified  by  transferring  the  origin  to  the  point 


©"■ 


this  first  set  of  axes. 


TEE   TETRAGONAL   SPACE-GROUPS   sl-Vfl. 


78 


TETRAGONAL  SYSTEM.^ 
A.  TETARTOHEDRY  OF  THE  SECOND  SORT. 


Space-Group  S\. 

One  equivalent  position : 

(a)  0  0  0.            (b)  00^ 

(c)  i 

|0. 

Two  equivalent  positions: 

(e)  OOu;    OOu. 

(f)  Hu;    Ha. 

(g) 

0|u; 

ion. 

Four  equivalent  positions: 
(h)  xyz;    yxz;    xyz; 

yxz. 

Space-Group  Si 

Two  equivalent  positions: 
(a)  0  0  0;    Hi 
(b)OOi;    HO. 

(c) 
(d) 

OH; 

lof 
OH 

(d) 


i  1  i 

3  3?. 


Four  equivalent  positions : 

(e)  OOu;    OOu;    |,  i  u-fi;    §,  h  i-u. 

(f)  0|u;    Hu;    h  0,  u+h;    0,  i  |-u. 

Eight  equivalent  positions: 

(g)  xyz;        yxz;        xyz;        yxz; 

x+5,  y+i  z-f-i;    i-y,  x-H,  |-z; 


l-x,  Hy,  z+i; 
y+i  5-x,  h-z- 


B.  HEMIHEDRY  OF  THE  SECOND  SORT. 


Space-Group  VJ. 

One  equivalent  position : 


(a)  0  0  0. 

(b)  Hi 

(c)  0  0^ 

(d)  h  h  0. 

Two  equivalent  positions: 

(e)  |0  0;    OH- 

(g)  OOu;    OOu. 

(f)  OH;   Hi 

(h)Hu;    HQ. 

Four  equivalent  positions: 

(i)   uOO 

;    uOO;    OuO 

OQO. 

(j)  uH 

;   uH;   Hi 

Hi 

(k)  u  0  ^ 

,    QO^;    Ou^ 

Otii 

(1)   uH 

uH;   HO 

HO. 

(m)  0  H 

;    OH;    Hu 

Hu. 

(n)  u  u  V, 

u  u  v;    uu  V 

utiv. 

^  Of  the  tetragonal  space-groups  those  marked  with  an  asterisk  will  be  found  to  have  co- 
ordinates differing  from  the  definitions  previously  given.  These  differences,  which  arise  from 
changes  of  origin,  have  been  introduced  to  bring  about  agreement  with  the  descriptions  of 
Niggli  (op.  cit.). 


74  THE   TETRAGONAL   SPACE-GROUPS   Vi-Vfl. 


Space-Group  VJ  (continued). 

Eight  equivalent  positions: 

(o)  xyz;    xyz;    xyz;    xyz; 

yxz;    yxz;    yxz;    yxz. 

Space-Group  Vj. 

Two  equivalent  positions: 

(a)  000;    OOi                (d)  0^0; 

§0i 

(b)nO;    OH-               (e)  OOi; 

OOf. 

(c)  IH;   HO.            (f)  Hi; 

Hi 

Four  equivalent  positions : 

(g)  uOO;    uOO;    Ou§;    Oui 

(h)  uH;    uH;    Iu0;    ^uO. 

(i)   u|0;    u^O;    i-u|;    ^ui 

(j)   uO^;    uO^;    OuO;    OuO. 

(k)  OOu;    OOu;    0,  0,  u+|;    0, 

0,  l-u. 

(1)   Hu;    Hu;    i  i  u-h^;    ^, 

i  ^-u. 

(m)0§u;    OH;    i  0,  u+i;    i 

0,  i-u. 

J&tfl'Af  equivalent  positions: 

(n)  xyz;               xyz;               xyz; 

xyz; 

y,  X,  z-f-l;    y,  X,  l-z;    y,  x, 

|-z;    y,  X,  z+i 

Space-Group  Va. 

Two  equivalent  positions: 

(a)  000;    HO.  (c)  0  §  u;    iOa. 

(b)00|;    Hi 
Four  equivalent  positions: 

(d)  OOu;    OOu;    Hu;    H^- 

(e)  u,  |-u,  v;    |-u,  u,  v;    u,  u+J,  v;    u-}-§,  u,  v. 

Eight  equivalent  positions : 

(f)  xyz;  yxz;  xyz;  yxz; 

i-x,  y-Fi,  z;    l-y,  ^-x,  z;     x-hi  ^-y,  z;    y+i  x-^^,  z. 
Space-Group  V^. 

Ti^o  equivalent  positions: 

(a)  000;    Hi  (b)  00|;    HO. 

Four  equivalent  positions: 

(c)  OOu;    OOu;    i  i  ^-u;    i  i  uH-i. 
(d)OH;    §0u;    i  0,  ^-u;    0,  i  u+i 

J5igf/i<  equivalent  positions : 

(e)  xyz;  yxz;  xyz;  yxz; 

l-x,  y+i  §-z;     ^-y,  ^-x,  z-f-|;     x-|-|,  §-y,  ^-z; 

y+i  x+i  z+i 


THE   TETRAGONAL   SPACE-GROUPS   yI-yI. 


75 


Space-Group  V^. 

Two  equivalent  positions: 

(a)  0  0  0;    HO. 

(b)  i  0  0;    0  A  0. 

Four  equivalent  positions: 


(c)  OH;    loh 

(d)  hhh   ooi 


(e)  OOu;    OOu 

1    1   ,T  .         11,, 

(f)   0|u;    OiQ 

1  0  u;    J  0  u. 

(g)  Hu;    ifu 

;    flu;    Hu. 

t  equivalent  positions : 

(h)  uOO;    OuO 

i  u+i  0; 

u+i 

i  0; 

uOO;    OuO 

h  l-u,  0; 

i-u, 

f,  0. 

(i)   uH;    |u| 

0,  u+i  i; 

u+i 

0,  i; 

,T    1    1  .         1   Vi    1 
U  2   2;        2^2 

0,  l-u,  i; 

l-u, 

0,  |. 

(j)   uuv;    tiuv, 

^-u,  u+l,  v; 

u+i 

u+i 

v; 

uuv;    uuv 

l-u,  §-u,  v; 

u+i 

i-u, 

V. 

(k)  u,  u+^,  v;    u,  u+^,  v;    u+|,  u,  v;    u+^,  u,  v; 
u,  ^-u,  v;    u,  |-u,  v;    |-u,  u,  v;    |-u,  u,  v. 

Sixteen  equivalent  positions: 

(1)    xyz;        xyz;  xyz;        xyz; 

yxz;        yxz;  yxz;        yxz; 

x+l,  y-l-iz;  x+l,  |-y,  z;    |-x,  y+|,  z;    |-x,  ^-y,  z; 

y+i  x+l,  z;  i-y,  x-|-|,  z;    y+i  |-x,  z;    |-y,  §-x,  z. 


Space-Group  V^. 

Foitr  equivalent  posit] 

ons: 

(a)  0  0  0, 

OOi; 

HO;    H 

1 

2- 

(b)  i  0  0, 

OH; 

OH;   H 

1- 

(c)  OOi, 

OOf; 

111.    11 

2    2    4;       11 

3 
4- 

(d)OH, 

OH; 

HI;   H 

i. 

Eight  equivalent  positions : 

(e)  uOO 

Ou^ 

u+i  i  0 

1 

>        2} 

u+l,  L- 

uOO 

Ou^ 

Hu,  i  0 

1 
2> 

i  — 11    i 

2         ^}     2' 

(f)  uH 

hnO 

u+i,  0,  i 

0, 

u+l,  0; 

Qii 

HO 

Hu,  0,  1 

0, 

Hu,  0. 

(g)  OOu 

Hu 

1     1     n-Ll 
2>    2j    U-t-2 

0, 

0,  u+i; 

OOti 

HQ 

h  h  Hu 

,    0, 

0,  Hu. 

(h)  0  A  u 

Hu 

i  0,  u+^ 

0, 

i  u+l; 

OH 

Hti 

i>  0,  Hu 

0, 

1,  Hu. 

(i)   Hu 

iH 

1       3       1_„ 

4)     4>     2        U 

.     1 

f        4> 

i  u+i; 

flu 

HQ 

i  h  Hu 

1, 

i  u+i 

76  THE   TETRAGONAL   SPACE-GROUPS   ¥^-¥4. 

Space-Group  V®  (continued). 
Sixteen  equivalent  positions: 

(j)   xyz;  xyz;  xyz;  xyz; 

y,  X,  z+l;        y,  X,  |-z;        y,  x,  ^-z;        y,  x,  z+|; 
x+i  y+i  z;    x+l,  l-y,  z;   l-x,  y+i  z;    ^-x,  |-y,  z; 
y+i  x+i  z+i;    l-y,  x+^,  ^-z;    y+|,  ^-x,  |-z; 

2~y>    1~X,    Z+2. 

Space-Group  "V^* 

Four  equivalent  positions : 


(a)  0  0  0; 

iOO; 

-HO; 

0^0. 

(b)HI; 

OH; 

OOi; 

|0i. 

(c)  HO; 

ifO; 

HO; 

HO. 

(d)H^; 

13    1. 

3    3    1. 

3    11 

4   4   2; 

4   4   2; 

4  4   2* 

Eight  equivalent  positions: 

(e)  OOu; 

OOu; 

|0u; 

§0U; 

r 

Hu; 

Hu; 

O^u; 

Olu. 

(f)  Hu; 

Hu; 

Hu; 

Hu; 

» 

HQ; 

Hu; 

HQ; 

Hu. 

(g)  iuO; 

ufO; 

f 

u  0;             u 

K 

3; 

l-u, 

i  0;    1 

'i+lj  i) 

.  0;    i 

u+i  0;     i 

1 
2 

-u,  0. 

(h)iu|; 

uH; 

fu|; 

U 

H; 

l-u, 

3    1. 

4j    2) 

u+l, 

hh 

i  u+i  i; 

1 
4j 

i-u,  i 

Sixteen  equivalent  positions: 

(i)    xyz; 

yxz; 

xyz; 

yxz; 

^-x, 

y,  z; 

l-y, 

X,  z; 

x+i  y,  z 

> 

y+i  X,  z; 

x+i 

y+i  z; 

y+i, 

l-x,  z 

;   l-x,  l-y 

,  z 

;   i-y,  x-Fi  z; 

X,  y+i  z; 

y,  1- 

X,  z; 

X,  h-y,  z 

> 

y,  x+i  z. 

Space-Group  V^* 

Four  equivalent  positions: 

(a)  0  0  0; 

i  0  1; 

HO; 

OH. 

(b)  HI; 

0§0; 

00|; 

10  0. 

(c)  Hi; 

HI; 

fH; 

IH- 

(d)iH; 

HI; 

HI; 

Hi 

£^ifif/i<  equivalent  positions : 

(e)  OOu; 

OOu; 

i  0, 

l-u; 

i  0,  u+l; 

Hu; 

1  1  ii . 
^  2  u, 

0,  i 

l-u; 

0,  i  u+|. 

(f)   iui; 

uHi 

1 

ftH; 

u 

H; 

l-u, 

3    3 . 

4»     4> 

u+i 

if; 

i  u-Fi  i; 

1 

4, 

l-u,  i 

(g)  HI; 

uH; 

fuf; 

u 

H; 

l-u, 

3      1. 

4)     4  > 

u+i 

i  i 

i  u-f-i  1; 

1 

4> 

l-u,  i 

(h)i|u; 

HQ; 

Hu; 

f 

iu; 

hh 

l-u; 

i,  1, 

u+l; 

1    3    i_„. 

4>     4j     2        "j 

f, 

i  u-hi 

THE   TETRAGONAL   SPACE-GROUPS   Vfl-Va. 


77 


Space-Group  V^  {continued). 
Sixteen  equivalent  positions: 


(i)    xyz;                    yxz; 

xyz;                  yxz; 

|-x,  y,  |-z;    |-y, 
x+i  y+i  z;    y+i 
X,  y+i  -2--z;    y,  1- 

X,  Z  +  2  j 

2     X,  z; 

■X,  z+i; 

x+i  y,  l-z;    y+i  x,  z+|; 
l-x,  i-y,  z;    i-y,  x+iz; 
X,  §-y,  l-z;    y,  x+iz+|. 

Space-Grou?  Vd. 

Four  equivalent  positions : 

(a)  0  0  0;    HO;    hOh 

(b)  ^0  0;    0§0;    0  0^; 
rp^lll.    111.    111. 

\.W444)        444>        444> 

(d)iil;   Hi;   fH; 

OH. 

HI. 

3    3    1 

444. 

Iff. 

Eight  equivalent  positions: 


(e)  OOu;    Hu;    I,  0,  u+|;  0,  |,  u+A; 

I,  0,  1— u;  0,  I,  |— u. 

Ill        11.  11  n_Ll. 

4;     4>     2        ">  4>     4>  ""r^> 


OOu;  Hu;  |,  0,  ?— u;  0,  |,  |-u. 

(f)   Hu;  Hu 

33,,.  31,-.,  3       3       l_n.  3  1  ,,_Ll 

4   4   U,  4   4   U,  4,     4,     2        ">  4>  4>  u   rj. 


;   u 

1 
2 

1. 

2  1 

;    u 

1 

h 

1 

>     2 

u 

1 . 

2  J 

;   1 

u 

1; 

;    u 

1 

4 

3 . 

4> 

J    u 

3 

4 

1. 

4> 

;   i 

u 

1; 

;   f 

u 

1 . 

4  ; 

;   u 

+ 

1 
2> 

;    u 

+ 

h 

1 

>     2 

— 

u, 

;   1 

- 

u, 

u+l,  i 

0; 

u+i  0,  ^; 

l-u,  i 

0; 

l-u,  0,  1; 

i  u-H, 

0; 

0,  u+i  1; 

i  i-u, 

0; 

0,  l-u,  i 

i  i-u, 

1 . 

4> 

u+i  i  f; 

l_n     i 
2        "j    4> 

1  . 

4; 

u+i  i  i; 

l-u,  i 

f; 

1,  u+i  f; 

3       1,, 
4)     2~U, 

3 . 

4; 

i  u+i  i. 

u+i  v; 

U 

■H,  u,  v+l; 

u, 

u+i 

v+i- 

5-u,  v; 

U 

-H,  u,  l-v; 

u, 

§-u, 

l-v; 

u+i  v; 

1 
2  ■ 

-u,  u,  |-v; 

u, 

u+i 

l-v; 

l-u,  v; 

h 

-u,  u,  v+i; 

u, 

i-u, 

v+i 

Sixteen  equivalent  positions: 
(g)  uOO 

uOO 

OuO 

OtiO 
(h)  iui 

uH 

ufi 
1 11 1 

4  u  4 

(i)    u  u  v 

UU  V 

uuv 

uti  V 
Thirty-two  equivalent  positions: 

(J)   xyz;        xyz;  xyz;        xyz; 

yxz;        yxz;  yxz;        yxz; 

x+iy+iz;  x+il-y,  z;    |-x,  y+i  z; 

y+i  x+i  z;  |-y,  x+iz;    y+i  f-x,  z; 

x+iy,  z+i-  x+i  y,  A- z;    |-x,  y,  |-z; 

y+2,  X,  z+2;  2~y>  X,  2~z;    y+2;  X,  2~z; 

X,  y+iz+l;  X,  l-yH-z;    x,  y+i  |-z; 

y,  x+i  z+i  y,  x+i  \-z;    y,  |-x,  |-z; 
Space-Group  V^". 

Eight  equivalent  positions: 

(a)  0  0  0;    0  01;  HO;    hh\. 

iOi-    iOO;  OH;    0|0. 


2~x,  2~y>  z; 
2~y>  2~"X,  z; 
5^~x,  y,  z+2; 
§-y,  X,  z+l; 

X,  h-Y,  z+i- 
y,  §-x,  z+i 


78 


THE   TETRAGONAL   SPACE-GROUPS   Vd°-Vd. 


Space-Group  V^^  (continued). 


(b)0  0|; 

OOf; 

H 

rl 

1  1 

2  2 

3. 
4; 

-|0f; 

|0i; 

oi 

3 

4 

Oi 

h 

(c)  iiO; 

1    3  n. 
4   4  ^> 

3  1 

4  4 

0 

3  3 

4  4 

3; 

111. 

4    4    2; 

3  11. 

4  4    2; 

1    3 
4    4 

1 

H 

1 

2- 

(d)HI; 

13    3. 

4    4    4) 

3  1 

4  4 

3 

4) 

3  3 

4  "4 

1. 

4; 

113. 
4    4    47 

3  11. 

4  4   4  J 

1    3 
4    4 

1 
4 

3  3 

4  4 

f. 

?en  equivalent  positions : 

(e)  uOO; 

Oiil; 

0, 

u+l,  0, 

u+l, 

i  0; 

uOO; 

Ou|; 

0, 

1 

2" 

-u,  0 

¥-u, 

i  0; 

11  i  i- 

"22) 

^uO; 

1 

2) 

u+i  1 

u+f, 

0,  ^; 

nil. 
U  2    2> 

iQO; 

1 
2j 

1 

2  " 

-u,  i 

i-u, 

0,  i 

(f)   OOu; 

O^u; 

0, 

1 
2> 

u-M 

0,  0, 

u+§; 

OOu; 

0|u; 

0, 

1 
2> 

i-u 

0,  0, 

l-u; 

Hu; 

iOu; 

h 

0, 

u-^i 

1    1 

2)     2j 

u+^; 

1  1  f, . 

2    2   "> 

|0u; 

1 

2) 

0, 

i-u 

i  i 

i--u. 

(g)  Hu; 

Hu; 

1 

4> 

1 

4> 

u-hl 

.        1       3 
4;     47 

u+l; 

iiti; 

13,-,. 
4    4   U, 

1 
4> 

1 
4> 

i-u 

3       1 
4}     4} 

l-u; 

3  3,,. 

4  4  U, 

3  1,,. 

4  4   U, 

3 

4; 

3 

4> 

u-hl 

!' !' 

u+^; 

ffti; 

3  1   fi  . 

4  4  U, 

3 

4) 

3 

4; 

l-u 

4>    4; 

i-u. 

(h)  iui; 

n   1   1  • 
U  4    4, 

1 
4> 

1 
2  ■ 

-u,  i 

u+i 

i  h 

1   f]    3  . 
4   U  4, 

11  i  ^• 
U  4    4, 

1 

4; 

u+l,  f 

l-u, 

1       3. 

4>     4; 

3  fi    1  . 

4  U  4, 

1,    3    1. 
U  4    4, 

3 

4) 

u+i  i 

l-u, 

3       1. 

4j     4> 

3  „    3, 

4  U  4, 

f,    3    3  . 

U  4   4, 

3 

4; 

1 

2  " 

-11     s. 

U,     4 

u+i 

f,    I. 

Thirty-two  equivalent  positions : 

(i)    xyz;  xyz;  xyz;  xyz; 

y,  X,  z+^;        y,  x,  ^-z;  y,  x,  |-z;  y,  x,  z-|-^; 

x+i  y+i  2;    x-f^,  i-y,  z;  ^-x,  y-fi  z;  |-x,  |-y,  z; 

y+2>    X-1-2)    Z+2>        2~y>    X-}-2,     2~Z;       y-r2>     2~X,     2~^'> 

2~y>  2~x,  z-f-j; 
x+i  y,  55-f-l;    x+l,  y,  l-z 
y-l-i  X,  z;        |-y,  x,  z; 


2     X, 


y,  i 


y+i  X,  z; 


i-x,  y,  z-l-l; 
l-y,  X,  z; 


X,  y+i  z-l-l;    X,  |-y,  |-z;    x,  y+|,  |-z;      x,  |-y,  z+^; 
y,  x-l-l,  z;         y,  x-f-|,  z;  y,  ^-x,  z; 


y,   §  — x,  z. 


Space-Group  V^.* 

Two  equivalent  positions : 

(a)  0  0  0;    H  i 
Fowr  equivalent  positions: 


(e)  100;    OH; 

(d)OH-;   OH; 

(e)  OOu;    OOu; 


OH; 


(b)00§;    HO. 
iOf. 


1 

2) 


u-H; 


1 

2} 


Eight  equivalent  positions: 


(f)   uOO;    OuO;    u+i  i  |;    i  u+|,  1; 
uOO;    OaO;    ^— u,  §,  1;    1,  |-u,  §. 


THE   TETRAGONAL   SPACE-GROUPS   ¥^-€4.  79 


(g)  uO| 

(h)  0  I  u 
0|u 

(i)  u  u  V 
tiu  V 


Space-Group  Vd^  (continued). 

Oui;    u+i  i,  0;  ^  u-f-i  0; 

Ou|;    A-u,  I,  0;  i  |-u,  0. 

|0u;    0,  h  u+l;  I,  0,  u+i; 

|0u;     0,  i  l-u;  i  0,  i— u. 

uuv;    u+i,  u+i  v+l;    u+i  ^-u,  |-v; 
uuv;    ^-u,  l-u,  v+l;    |-u,  u+|,  |-v. 
Sixteen  equivalent  positions: 

(J)  xyz;  xyz;  xyz;  xyz; 
yxz;  yxz;  yxz;  yxz; 
x+i  y+i  z+i;    x+i  |-y,  ^-z;    |-x,  y+|,  ^-z; 

§-x,  i-y,  z+^; 
y+i  x+i  z+l;    |-y,  x+|,  |-z;    y+|,  ^-x,  |-z; 

5-y,  l-x,  z+i 
Space-Group  V^  *• 

Four  equivalent  positions: 

(a)  000;    iOi;    HI;    OH- 


(b)00|;    |0|;    HO;    OH- 

i  equivalent  positions 

(c)  OOu;             OOu;              i  0,  i-u; 

i  0,  u+i; 

i  i  u-hl;    1,  i  l-u;    0,  |,  f-u; 

0,  i  u+i 

(d)|u|;             uH;             fui; 

ail; 

i  u+i,  f;    u+l,  i  f;    I  l-u,  f; 

l-u,  f,  f. 

Sixteen  equivalent  positions: 

(e)  xyz;  yxz;  xyz;  yxz; 

l-x,  y,  i-z;    l-y,  x,  z+J;     x-hi  y,  i-z;    y+i  x,  z+i; 
x+i  y+i  z-l-l;    y-t-i  |-x,  |-z;    |-x,  |-y,  z+|; 

l-y,  x+i  I-z; 
X,  y+l,  f-z;    y,  |-x,  z+f ;    x,  |-y,  |-z;    y,  x-M,  z+f. 

C.  TETARTOHEDRY. 
Space-Group  CJ. 

One  equivalent  position : 

(a)  OOu.  (b)  Hu. 

Ttyo  equivalent  positions: 

(c)  0|u;    |0u. 
Four  equivalent  positions: 

(d)  xyz;    yxz;    xyz;    yxz. 
Space-Group  Ci 

Four  equivalent  positions: 

(a)  xyz;    y,  x,  z+l;    x,  y,  z+|;    y,  x,  z-|-f. 


80  TEE   TETRAGONAL   SPACE-GROUPS   C4-C^. 

Space-Group  Ct 

Two  equivalent  positions: 

(a)  OOu;    0,  0,  u+i  (c)  O^u;    I  0,  u+i 

(b)  H  u;    i  I,  u+i 

Four  equivalent  positions : 

(d)  xyz;    y,  x,  z+|;    xyz;    y,  x,  z+^. 
Space-Group  Ct 

Four  equivalent  positions : 

(a)  xyz;    y,  x,  z+|;    x,  y,  z+^;    y,  x,  z+i. 
Space-Group  C|. 

Two  equivalent  positions: 

(a)  OOu;    i  i,  u+i 
Four  equivalent  positions: 

(b)  Olu;    iOu;    i  0,  u+^;    0,  i  u-^i 
Eight  equivalent  positions: 

(c)  xyz;        yxz;        xyz;        yxz; 

x+2>  y+2>  z+2;    2~y>  x+2,  z+2;    2~x,  2~y)  z4-2j 

y"r2>    2~X,    z+2' 

Space-Group  C^. 

/^owr  equivalent  positions : 

(a)  OOu;    0,  i  u+i;    |,  0,  u+|;    i  i  u+i 
^zgi/if  equivalent  positions: 

(b)  xyz;    y,  ^-x,  z+|;    xyz;    |-y,  x,  z+f; 

x+2>  y"F2>  z+2;   y+2>  X,  z+j;    2~x,  2~y>  z+2; 

y,  x+i  z+i. 

D.  PARAMORPHIC  HEMIHEDRY. 
Space-Group  04^,. 

One  equivalent  position: 

(a)  000.  (b)  00|.  (c)  HO.  (d)  Hi 

Two  equivalent  positions: 

(e)  0|0;    ^0  0.  (g)  OOu;    OOu. 

(f)  OH;    ^0|.  (h)  Hu;    HQ. 
Four  equivalent  positions: 

(i)  O^u;  iOu;  0|u;  ^Ou. 
(j)  uvO;  vuO;  uvO;  v  u  0. 
(k)  uv|;    vu|;    uv|;    vu^. 


THE   TETRAGONAL   SPACE-GROUPS    cA-Cil,.  81 

Space-Group  C4h  (continued). 
Eight  equivalent  positions : 

(1)    xyz;    yxz;    xyz;    yxz; 
xyz;    yxz;    xyz;    yxz. 

Space-Group  C^. 

Two  equivalent  positions: 

(a)  0  0  0;    0  0  i.  (d)  OH;    ^0  0. 

(b)HO;    HI.  (e)  OOi;    0  0  f . 

(c)  0^0;   |0i  (f)  Hi;   Hi 

Four  equivalent  positions : 

(g)  OOu;    OOu;    0,  0,  u-}-|;    0,  0,  |-u. 

(h)    2    2  U;        2   2  U;        2]     2)    U"l    2;        2)     2>     2~U. 

(i)    Olu;    Oiu;    i  0,  u4-i;    i  0,  ^-u. 
(j)   uvO;    uvO;    vu|;    vuf. 
E'zgf/if  equivalent  positions : 

(k)  xyz;    y,  x,  z-hi;    xyz;    y,  x,  z-Hi; 
xyz;    y,  x,  Hz;    xyz;    y,  x,  ^-z. 

Space-Group  CA. 

Two  equivalent  positions: 

(a)  010;    100.        (b)0i-3;    1 0  i.        (c)OOu;    H  u. 
Four  equivalent  positions : 

(d)  HO;    fiO;    HO;    ffO. 

fa\     ill.lli»131.331 
\pj     442>        442>        i   4   2  )       442* 

(f)  O^u;    |0u;    |0u;    0  H- 
iJzgf^i  equivalent  positions: 

(g)  xyz;  yxz;  xyz;  yxz; 

Hx,  Hy,  z;    y+i  i-x,  z;    x+|,  y-l-i  z;    Hy,  x+|,  z. 
Space-Gi  cup  C/h. 

Tiwo  equivalent  positions: 

(a)  OH;   |of.  (b)  OH;   ioi 

Four  equivalent  positions : 

/'p^ll^J•     311.     33n.     131 

V  W     44^>        442;        44'-'>        442- 
\M)    442>        44"-'>        442>        44'-'' 

(e)  OOu;    Hu;    i  i  |-u;    0,  0,  u+i 

(f)  O^u;    iOu;    i  0,  u-hi;    0,  |,  Hu. 
Eight  equivalent  positions: 

(g)  xyz;  y,  x,  z-H;  xyz;  y,  x,  z-|-§; 
^-x,  Hy,  z;    y+i  i-x,  Hz;    x-Fi  y+|,  z; 


Hy,  x+l, 


i_. 


82 


THE  TETEAGONAL  SPACE-GROUPS  C^-C^- 


Space-Group  C^. 


Two  equivalent  positions: 

(a)  0  0  0;    -HI. 

(b)  0  0  1 

;    HO. 

Four  equivalent  positions: 

(c)  0|0;     |0  0;    i  0 1, 

OH. 

(d)OH;   Hi;   OH; 

lOf. 

(e)  OOu;    OOu;    i  i 

u+l;    h 

i  l-u. 

Eight  equivalent  positions: 

(f)  iii;   fii;   ffi; 

ill; 

113,        133.        1    11. 

4   4  3>        4   4  2>        4   4   4> 

fH. 

(g)  0|u;    |0u;    i  0, 

u+l; 

0,  1,  u+l; 

Olu;    |0u:    i  0, 

l-u; 

0,  i  l-u. 

(h)  uvO;    vuO;    |— v, 

u+l,  1; 

u+i  y-\-h  h 

uvO;    vuO;    v+l, 

l-u,  1; 

l-u,  l-v,  |. 

Sixteen  equivalent  positions: 

(i)    xyz;        yxz;        xyz;        yxz; 
xyz;        yxz;        xyz;        yxz; 

x+i  y+l,  z+l;    l-y,  x-hi  z+l;    |-x,  |-y,  z+f; 

y+i  l-x,  z+l; 
l-x,  §-y,  |-z;    y+l,  |-x,  |-z;    x+|,  y+l,  |-z; 

l-y,  x-}-|,  \-z. 
Space-Group  C^.*  ♦ 

Four  equivalent  positions: 

(a\    nil.        nl'.        115.        133 
(\<\    n35.       nl3.        111.        137 


?  4   8- 


Eight  equivalent  positions: 


(c)  000;    iil;    |0|;    Hi; 

o|0;   Hi;   IH;   Hi 

(d)OOA;    iii;    ^00; 
HO; 


n  1  1. 


111. 

4   4   4> 
9   3  3.. 

4   4   4; 


13  1. 
4  ¥?» 
3  11 

744. 


(e)  Oiu;    i  i,  u+f;    i  f,  u+|; 


Of  u;    h,  f,  i-u;    i  i, 


1   i_ 


t-u; 


0,  f,  u+i; 
0,  i,  f-u. 


Sixteen  equivalent  positions: 

(f)   xyz;    y-f-i,  i-x,  z+f;    \-x,  y,  z-j-^; 
X,  y+i  z;    y-f-i  f-x,  i 


i_ 


z: 


I-x,  I 


x+l,  y+l,  z-l-l;    y+f,  f-x,  z-Hi;    x, 
x+l,  y,  l-z;    y+f,  i-x,  f-z;    xyz; 


i-y,  x-f-f,  z-f-i; 

-y,  l-z; 

i-y,  x-l-i,  f-z  ; 
l-y,  z; 
f-y,  x-F-  i,  z+f; 
f-y,  x+f,  i-z. 


THE   TETRAGONAL   SPACE-GROUPS   civ-Ci^v-  83 

E.  HEMIMORPHIC  HEMIHEDRY. 
Space-Group  C4V 

One  equivalent  position : 

(a)  0  0  u.  (b)  H  u. 

Two  equivalent  positions: 

(c)  0|u;    iOu. 
Four  equivalent  positions: 

(d)uuv;    uuv;    uuv;     uQv. 

(e)  uOv;     Ouv;    uOv;     Otiv. 

(f)  u^v;     |uv;    u  ^  v;     ^uv. 

Eight  equivalent  positions: 

(g)  xyz;    yxz;    xyz;    yxz; 
yxz;    xyz;    yxz;     xyz. 

Space-Group  C4V. 

Two  equivalent  positions : 

(a)  OOu;    Hu-  (b)  O^u;    |0u. 

Four  equivalent  positions: 

(c)  u,  |-u,  v;    u-M,  u,  v;    u,  u-M,  v;    |-u,  u,  v. 
Eight  equivalent  positions: 

(d)  xyz;  yxz;  xyz;  yxz; 

y+l,  x+l,  z;     x-f-i^-y,  z;    ^-y,  ^-x,  z;    ^-x,  y+i  z. 

Space-Group  Ct,. 

Two  equivalent  positions : 

(a)  OOu;    0,  0,  u-hi  (b)  H  u;    i  i  u-j-^ 

Four  equivalent  positions: 

(c)  Oiu;    iOu;     i  0,  u+l;    0,  i  u-|-i 

(d)  uuv;    uuv;    u,  u,  v-f-^;    u,  u,  v-|-|. 

Eight  equivalent  positions: 

(e)  xyz;    y,  x,  z+|;     xyz;     y,  x,  z-\-\; 
yxz;     X,  y,  z-fl;    yxz;    x,  y,  z-f-§. 

Space-Group  C4V.* 

Two  equivalent  positions: 

(a)  OOu;    h  h  u+|. 
Four  equivalent  positions: 

(b)0|u;    0,  i  u+i;  ^Ou;     i  0,  u-^. 

(c)  uuv;    u+l,  |-u,  v-f|;    uuv;    ^-u,  u-M,  v-|-^. 


84  THE   TETRAGONAL   SPACE-GROUPS   C4V-C4V. 

Space-Group  C^y  {continued). 
Eight  equivalent  positions : 

(d)  xyz;     y+i  |-x,  z+|;     xyz;     |-y,  x+|,  z+|; 
l-x,  y+i  z+^;    yxz;    x-Hl,  |-y,  z4-^;    yxz. 

Space-Group  C/v 

Two  equivalent  positions : 

(a)  OOu;    0,  0,  u+i  (b)Hu;    i  §,  u+i 

Fowr  equivalent  positions : 

(c)  O^u;    iOu;     i  0,  u+|;    0,  |,  u-^. 
^tgf/i<  equivalent  positions: 

(d)  xyz;  yxz;  xyz;  yxz; 

y,  X,  z+§;     X,  y,  z+|;     y,  x,  z+i;     x,  y,  z+|. 

Space-Group  C/y. 

Two  equivalent  positions: 

(a)  OOu;    i  I,  u+i 
i^owr  equivalent  positions: 

(b)0|u;     iOu;     i  0,  u+|;    0,  i  u+i 
^tg/if  equivalent  positions : 

(c)  xyz;  yxz;  xyz;  yxz; 

y   I    2>    X+2)    Z-t-j-;       X+2)     2~y>    ^12)        2~y)     2~X,    Z+2J 

2~x,  y-|-2>  z+2. 
Space-Group  C/y. 

Tw/'o  equivalent  positions : 

(a)  OOu;    0,  0,  u+|.  (b)Hu;    i  i  u+i 

(c)  Oiu;    i  0,  u-hi 
Four  equivalent  positions : 

(d)uOv;    uOv;    0,  u,  v+l;    0,  u,  v-hi 

(e)  u^v;    u|v;     i  u,  v-f-^;     i  u,  v-\-^. 
Eight  equivalent  positions: 

(f)  xyz;  y,  x,  z+^;    xyz;  y,  x,  z-f-^; 
y,  X,  z+l;    xyz;                y,  x,  z-|-^;    xyz. 

Space-Group  C^y. 

Four  equivalent  positions : 

(a)  OOu;     Hu;    i  i  u-^;    0,  0,  u-hi 
(b)0|u;    ^Ou;    i  0,  u+f;    0,  i  u-l-|. 
Eight  equivalent  positions: 

(c)  xyz;  y,  x,  z-}-|;  xyz;  y,  x,  z+i; 

y+i  x+l,  z+l;    x4-|,  |-y,  z;    ^-y,  |-x,  z-|-|; 

^-x,  y+i,  z. 


THE   TETKAGONAL   SPACE-GROUPS   Ciy-Cli-  85 

Space-Group  C'^. 

Two  equivalent  positions: 

(a)  OOu;    i,  i  u+i 
Four  equivalent  positions : 

(b)Oiu;    lOu;    i  0,  u+^;    0,  i  u+i 
Eight  equivalent  positions : 


(c)  u  u  V 
uti  V 

(d)  u  0  V 
tiOv 


uuv;  u+i  ^-u,  v+^;  u+i  u+|,  v+^; 

uuv;  ^-u,  u+l,  v4-^;  i-u,  |-u,  v-|-^. 

Ouv;  I,  u+l,  v+^;  \\-\-\,  i  v+|; 

Ouv;  i  ^-u,  v+^;  |-u,  i  v+^. 


Sixteen  equivalent  positions: 

(e)  xyz;    yxz;     xyz;     yxz; 
yxz;    xyz;    yxz;    xyz; 

x+i  y+i  z+^;     |-y,  x+|,  z+^;    |-x,  ^-y,  z+^; 

y+i  l-x,  z-l-l; 
y+i  x+i  z+^;    x+i  l-y,  z4-^;    |-y,  ^-x,  z-h|; 

l-x,  y-l-i  z+\. 

Space-Group  Cl". 

Four  equivalent  positions: 

(a)  OOu;    Hu;    \,  h,  n+h    0,  0,  u-Hi 
(b)Oiu;    iOu;    i  0,  u-f-i;    0,  i  u+i 

^tgr/i<  equivalent  positions: 

(c)  u,  u+i  v;    ^-u,  u,  v;    %  u+^,  v+|;    u-f-|,  u,  v+^; 
u,  ^-u,  v;    \i-\-\,  u,  v;    u,  ^-u,  v+^;    ^-u,  u,  v-}-^. 

Sixteen  equivalent  positions: 

(d)  xyz;  yxz;  xyz;  yxz; 

y,  X,  z-f^;    x,  y,  z-F-|;    y,  x,  z+\;    x,  y,  z-\-\', 
x+l,  y+l,  z-f^;    \-y,  x-f-|,  z-fl;    |-x,  ^-y,  z-j-|; 

y+i  l-x,  z-M; 
y+i  x-fi  z;    x-l-i  |-y,  z;    |-y,  |-x,  z;    |-x,  y+i  z. 

Space-Group  Civ-* 

i^ow?'  equivalent  positions: 

(a)  OOu;    0,  i,  u-hi;    i  0,  u+|;    i  i  u+i 
Eight  equivalent  positions: 

(b)  Ouv;    u,  i  v-f-J;    i  u+|,  v-f-^;    |-u,  0,  v-f-f; 
Ouv;    %  i  v+i;    i  |-u,  v-|-|;    u-|-i  0,  v+|. 


86 


THE   TETRAGONAL   SPACE-GROUPS   C\;-T>1. 


Space-Group  Cl*  (continued). 
Sixteen  equivalent  positions: 

(c)  xyz;  y,  ^-x,  z+i;  xyz;  |-y,  x,  z+f; 
xyz;  y,  |-x,  z+J;  xyz;  y+i  x,  z+f; 
x+i  y+l,  z+§;     y+i  X,  z+f;     |-x,  §— y,  z+|; 


2~x,  y+2>  z+^;    ^• 

Space-Group  CH* 

Eight  equivalent  positions: 


■y,  X,  z+f; 


y,  x+i  z+^; 
x+i  |-y,  z+l; 

y,  x+i  z+i 


(a)  OOu;    0,  i  u+i;    |,  0,  u+f;    0,  0,  u+i; 
Hu;    I,  0,  u+i;    0,  I,  u+f;    i  i,  u+i 

Sixteen  equivalent  positions : 

(b)  xyz;  y,  |-x,  z+|;  xyz;  §-y,  x,  z+f; 
X,  y,  z+l;  y,  l-x,  z+f;  x,  y,  z+§;  y+|,  x,  z+f; 
x+l,  y+l,  z+l;     y+l,  x,  z+f;     §-x,  f-y,  z+|; 

y,  x+i  z+f; 
l-x,  y+i  z;    i-y,  X,  z+f;    x+i  §-y,  z;    y,  x+|,  z+f. 

F.  ENANTIOMORPHIC  HEMIHEDRY. 

Space-Group  D4. 

One  equivalent  position : 


(a)  0  0  0. 

(c)  H 

.0. 

(b)  0  0  |. 

(d)H 

h 

Two  equivalent  positions : 

(e)  OiO; 

§0  0. 

(g)  OOu;    OOu 

(f)  OH; 

1  n  i 

2  0  2- 

(h)Hu;    HG 

Four  equivalent  positions: 

(i)    Olu; 

§0u;    0|u 

§0u. 

(j)   uuO; 

uuO;    utiO 

;    uuO. 

(k)  uu§; 

uu|;    uu| 

uu|. 

(1)   uOO; 

OuO;    uOO 

OuO. 

(m)uH; 

§u|;    uH 

Hi 

(n)  uO§; 

Ou§;    uO§ 

oa|. 

(0)  uiO; 

|uO;    u|0; 

|uO. 

Eight  equivalent  positions: 

(P)  xyz; 

yxz;    xyz;    yj 

4z; 

yxz; 

xyz;    yxz;    x] 

^z. 

Space-Group  D^* 

Two  equivalent  positions: 

(a)  0  0  0;    H  0. 

(b)  001;    Hi 


(c)  0§u;    §0u. 


THE   TETRAGONAL  SPACE-GROUPS  DJ-D^. 


87 


Space-Group  D4  {continued). 
Four  equivalent  positions: 


1  1  r, . 

2  2  u, 


i  i  11 
2  2  Li* 


(d)  OOu;    OOu; 

(e)  uuO;    uuO;    u+^,  ^—\i,  0;    |— u,  u+|,  0. 

(f)  uu|;    ua|;    u+i  |-u,  |;    §-u,  u+i  f. 


^tgr/i^  equivalent  positions: 
(g)  xyz;    y+i  |-x,  z; 
^-x,  y+l,  z;    yxz; 

Space-Group  Df.* 

Four  equivalent  positions: 

(a)  OuO;    uOf;    Ou|;    u  0  i 

(b)  |ui;    uH;    IQO;    u  H- 


xyz;    l-y,  x+i  z; 

x+i  l-y,  z;     yxz. 


(c)  uuf; 


uu|;    uu|; 


uuf. 


Eight  equivalent  positions: 


(d)  xyz;    y,  x,  z+f;    x,  y,  z+|;    y,  x,  z+i; 
xyz;    y,  x,  i-z;    x,  y,  |-z;    y,  x 

Space-Group  Dt* 


l-z. 


Four  equivalent  positions: 

(a)  uuO;    uu^;    u+l,  ^-u,  f;    |-u,  u-|-|,  J. 


^ig^i  equivalent  positions: 

(b)  xyz;    y+^,  ^-x,  z+f; 
^-x,  y+i  i-z;    y,  X, 

Space-Group  Dt* 

Two  equivalent  positions: 


X,  5^,  z+l; 
x+i 


h-z 


-y,  x+l,  z+i; 

■y,  f-z;   yxz. 


(a)  0  0  0; 

ooi. 

(d)OH 

HO. 

(b)HO; 

Hi 

(e)  00  f; 

oof. 

(c)  0^0; 

^oi 

(f)  Hf; 

Hf. 

/^owr  equivale 

Qt  positions: 

(g)  OOu 

OOu; 

0,  0,  u-f-i;    0, 

0,  Hu. 

(h)  Hu 

HQ; 

i  i  u-}-^;    i 

i  5-u. 

(i)    0|u 

OH; 

1,  0,  u+l;    1, 

0,  l-u. 

(j)   uOO 

uOO; 

Ou|;    Oui 

(k)uH 

uH; 

HO;    1  u  0. 

(1)    uOi 

uO§; 

OuO;    OuO. 

(m)  u  1  0 

uiO; 

H^;    Hi 

(n)  u  u  i 

;    uuf; 

uuf;    uuf. 

(0)  uuf 

;    uuf; 

uuf;    uuf. 

Eight  equivale 

nt  positi 

ons: 

(P)  xyz; 

y,  X,  z+l;    xyz;    y,  x, 

z+i 

xyz; 

y,  X,  1- 

-z;     xyz;     y,  x, 

Hz. 

88  THE   TETRAGONAL   SPACE-GROUPS   D4-D4. 

Space-Group  Dt* 

Two  equivalent  positions; 

(a)  000;    HI-  (b)  00^;    HO. 

Four  equivalent  positions: 

(c)  OOu;  OOu;  h  h,  |-u;  h  i  u+h 

(d)Oiu;  |0u;  i  0,  |-u;  0,  i  u+i. 

(e)  uuO;  uuO;  u+|,  |-u,  |;  ^— u,  u+|,  | 

(f)  uu|;  uu|;  u+i  |-u,  0;  ^-u,  u+|,  0. 

Eight  equivalent  positions: 

(g)  xyz;    y+^,  |-x,  z+|;    xyz;    ^-y,  x+i,  z+|; 
§-x,  y+i  i-z;    yxz;    x+i  l-y,  |-z;    yxz. 

Space-Group  D4.* 

Four  equivalent  positions: 

(a)  OuO;    uOi;    Ou|;    uOf. 

(b)  |ui;    uH;    HO;    u  H- 

(c)  uu|;    uuf;    uuf;    u  u  |. 

Eight  equivalent  positions: 

(d)  xyz;     y,  x,  z+i;     x,  y,  z+|;    y,  x,  z-f-f; 
xyz;     y,  x,  f-z;    x,  y,  |-z;    y,  x,  |-z. 

Space-Group  Df.* 

Four  equivalent  positions: 

(a)  uuO;    util;    u-H,  |-u,  i;    |-u,  u-f-^  f. 
Eight  equivalent  positions : 

(b)  xyz;    y-f-^  |-x,  z-f-|;    x,  y,  z4-|;    |-y,  x+|,  z-f-|; 
yxz;    l-x,  y-f-j,  |-z;    y,  x,  |-z;    x-j-^  |-y,  i-z. 

Space-Group  D4. 

Ti^o  equivalent  positions: 

(a)  0  0  0;    HI-  (b)  0  0  J;    HO. 

Four  equivalent  positions: 

(c)  0^0;    1 0  0;    HI;    OH- 

(d)  OH;   hoh   HI;   OH- 

(e)  OOu;    OOu;    i  i  u-|-|;    |,  I,  |-u. 
Eight  equivalent  positions : 


(f) 

OH; 

OH; 

0,  h  u+l; 

0,  1, 

Hu; 

Hu; 

HQ; 

i  0,  u-fl; 

1,  0, 

Hu. 

(g) 

uuO; 

uuO; 

u-f-i  u-l-l,  1; 

Hu, 

u+l, 

1; 

utiO; 

utiO; 

Hu,  Hu,  1; 

u+i 

Hu, 

I. 

THE   TETRAGONAL   SPACE-GROUP  D4-D4. 


89 


Space-Group  D4  {continued), 

(h)  uOO;    OuO; 

tiOO; 

1 . 
5> 


OuO; 
Oui; 


It 


u+^, 

1 

1 . 

I' 

i 

u+i 

1 . 

i-u, 

1 

1 . 

2> 

i 

i-u, 

1 
I- 

u-hi 

i, 

0; 

i 

u+i 

0; 

l-u, 

i 

0; 

1 
1? 

l-u, 

0. 

(i) 

(j)  u,  u+i,  I;    u,  i-u,  J;    u+i  u,  f ;    i-u,  u,  |; 


^-u,  u, 


1 . 


U  +  ^ 


u,  i-u,  f ;    ii,  u-Hi  f. 


Sixteen  equivalent  positions: 

(k)  xyz;        yxz;        xyz;        yxz; 
yxz;        xyz;        yxz;        xyz; 

x+l  y+i  z-|-§;    |-y,  x+|,  z+|; 

y+2>  x-f-2,  2~z;    x-|-2>  2~y>  2~z; 

Space-Group  D\°.* 

i^our  equivalent  positions: 

(a)  000;    OH;    iOf;    Hi 

(b)  00^;    OH;    iOi;    HO. 
Etgf/i<  equivalent  positions: 


l-y,  z+^; 


y+i  Hx,  z-l-^; 
l-y,  i-x,  l-z; 

2~x,  y-|-^,  2~z. 


(c) 
(d) 
(e) 
(f) 


OOu 

0, 

OOu, 

0, 

uuO 

;   u, 

uuO 

;   u, 

uuO 

»   a, 

uuO 

u, 

uH, 

h 

GH 

3 

4? 

u+i 


u+l, 

Hu, 

u-hi 


1 

2) 

^>    4       U 

1 

1 
¥ 
1 

4 
1 

4 

a 

8 
3 

8 


h  0, 

u+f; 

1       1 

2>     2> 

Hu; 

io, 

l-u; 

hh 

u+i 

l-u, 

l-u, 

1  . 

l-u, 

u,  1; 

u-hi 

u+i 

1; 

u+i 

u,  f. 

u-f-i 

l-u, 

^; 

u+l, 

u,  1; 

Hu, 

u+l, 

1 . 

2) 

l-u, 

ti,  f. 

l-u, 

1       5. 

4)     8) 

iu|; 

u+i 

3       5. 

4;     87 

|a|. 

Sixteen  equivalent  positions: 
(g)  xyz; 


2     X, 


4        Z, 


y,  Hx,  z+l; 
2~y>  ^~x,  2~ 


x+i  y+i  z+l;    y+i  x,  z+f; 


X,  y+i  i-z; 


yxz; 


xyz;     Hy,  x,  z+|; 

x+i  y,  l-z;      yxz; 
l-x,  i-y,  z+l; 

y,  x+i  z+i; 
X,  |-y,  i-z; 

y+i  x+i  Hz. 


G.  HOLOHEDRY. 


Space-Group  DA 


One  equivalent  position: 
(a)  0  0  0.  (c) 

(b)OOi  (d) 

Two  equivalent  positions: 

(e)  OH;   io|. 

(f)  0^0;    ^00. 


Ill 


(g)  OOu; 
(h)Hu; 


OOu. 

1     1    vi 


90 


THE   TETRAGONAL   SPACE-GROUPS   B^-D^n. 


Space-Group  D^ 

{continued). 

Four  equivalent  positions: 

(i)  OH; 

|0u; 

Hu, 

OH. 

(j)   uuO 

uuO, 

uuO 

uuO. 

(k)  uui 

uu| 

uu| 

;    Qui 

(1)   uOO 

OuO; 

OuO 

uOO. 

(m)  u  0  1 

Ou|; 

Ou| 

uOi 

(n)  uH, 

iuO; 

|uO 

u|0. 

(o)  uH, 

|u|; 

huh 

uH. 

Eight  equivale 

nt  posit 

ions: 

(p)  uvO 

vuQ 

vuO 

;    uvO; 

vuO 

uvO 

uvO 

;    vuO. 

(q)  u  V 1 

vu| 

vu| 

;    uv^; 

vu| 

;  \uuv 
;    uuv 

u  V  ^ 

;    vui 

(r)  u  u  V 

;    uuv 

;    uuv; 

uuv 

;    uuv 

;    uuv. 

(s)  Ouv 

tiOv 

uOv 

;    Ouv; 

uOv 

Ouv 

Ouv 

;    uOv. 

(t)  |uv 

uiv 

u|  V 

;    Hv; 

U^  V 

iuv 

|uv 

;   u|v. 

Sixteen  equivalent  positions : 

(u)  xyz;    yxz;    xyz;    yxz; 

yxz;    xyz;    yxz;    xyz; 

xyz;    yxz;    xyz;    yxz; 

yxz;    5cyz;    yxz;     xyz. 

Space-Group  D^. 


iiO; 


(c)   f  f 

(d)Hi; 


hoh 


111 

^  2  !• 

113 

^2   4' 


Two  equivalent  positions: 

(a)  0  0  0;    0  0|. 

(b)  OOi;    oof. 
Four  equivalent  positions: 

(e)  OH;    ^00;    OH; 

(f)  OH;   Hi;   HI;   oH- 

(g)  OOu;    OOu;    0,  0,  Hu;    0,  0,  u+i 
(h)  Hu;    Hu;    i  i  l-u;    i  i  u+i 

Etgf/if  equivalent  positions : 


(i) 

OH, 

Hu; 

OH, 

HQ; 

(J) 

uuO 

;    UU 0; 

uuO 

;    QuO; 

(k) 

uOO, 

OuO; 

uOO; 

OuO; 

(1) 

uH; 

H§; 

QH, 

HI; 

(m) 

^  V  i, 

vui; 

uvi, 

vui; 

0,  i 

u+l;    h, 

0, 

u-f-l; 

0,  i 

Hu;    i, 

0, 

Hu. 

uu^l 

uQ§; 

Qui; 

Qui 

uOi; 

Ou|; 

QOi; 

OQi 

uH; 

HO; 

uH; 

HO. 

vuf; 

uvf; 

vQf; 

Qvf. 

THE   TETRAGONAL   SPACE-GROUPS   Biir^*^'  91 

Space-Group  D^  {continued). 
Sixteen  equivalent  positions : 
(n)  xyz;    yxz;    xyz;    yxz; 


yxz;    xyz;    yxz;    xyz; 

X,  y,  l-z;    y,  x,  ^-z;    x,  y, 

l-z;    y,  X,  i-z; 

y,  X,  z+l;    X,  y,  z+l;    y,  x, 

z+i;    X,  y,  z+i 

:e-Group  Dtt. 

Two  equivalent  positions : 

(a)  000;    HO.                (c)  0|0; 

100. 

(b)  001;     111.                (d)  OH; 

101. 

Four  equivalent  positions: 

(e)  HO;    HO;     i  f  0;    ff  0. 

(f)  iH;   fil;    iH;   HI- 

(g)  OOu;    OOu;    Hu;    H  u. 

(h)  0|u;    |0u;    ^Ou;    0|u. 

Eight  equivalent  positions: 

(i)   uuO;    uuO;    u-H,  u-h^,  0;  u+l,  i— u,  0: 

utiO;    tiuO;    §— u,  ^— u,  0;  |— u,  u+^,  0. 

(j)   uu|;    uu^;    u-f-^,  u+i  |;  u+i  |-u,  |; 


uu|; 

uu|; 

l-u, 

l-u,  1; 

l-u,  u+i 

i 

(k)  uOO; 

OuO; 

u+i 

h  0; 

i  u+i  0; 

uOO; 

OuO; 

^-u, 

i  0; 

i  ^-u,  0. 

(1)  uH; 

|u|; 

u+i 

0,  h 

0,  u+i  i; 

tiH; 

|u|; 

l-u, 

0,  1; 

0,  l-u,  i 

(m)u,  ^- 

■u,  v; 

u+i  u 

i,  v;    1- 

u,  u,  v;    u,  u 

i+i 

v; 

l-u, 

u,  v; 

u,  i-u 

i,  v;    u, 

u+i  v;    u+t 

^,  Q, 

V. 

Sixteen  equivalent  positions: 

(n)  xyz; 

yxz; 

xyz 

;      yxz; 

~ 

yxz; 

xyz; 

yxz 

;      xyz; 

^-x, 

?-y,  z; 

y+h 

§-x,  z; 

x+i  y+i  z; 

1 

-y,  x-hi  z; 

l-y, 

i-x,  z; 

^-x, 

y+i  z; 

y+h  x+i  z; 

x+i  5-y,  z. 

Space-Group  D^- 

Two  equivalent  positions: 

(a)  0  0  0; 

Hi 

(b)  OOi 

;   HO. 

Four  equivalent  positions: 

(c)  10  0; 

0|0; 

OH; 

^Oi 

(d)^Oi; 

OH; 

OH; 

lOf. 

(e)  OOu; 

OOu; 

hh 

i-u;    1, 

h  u+1. 

Eight  equivalent  positions : 

(f)  Hi; 

Uh 

iH; 

IH; 

iH; 

fH; 

Hf; 

Hi 

92 


THE   TETRAGONAL   SPACE-GROUPS   dA-d/i,. 


Space-Group  D4  (continued). 


(g) 

^Ou, 

O^u; 

^Oii, 

OiQ; 

(h) 

uuO 

;  uuO; 

uuO 

;  uu  0; 

(i) 

uOO 

OuO; 

uOO 

OuO; 

(3) 

uOl 

Ou^; 

uO| 

Oui; 

0,  i  u+i; 

1,  0, 

u+l; 

0,  i  ^-u; 

h  0, 

i-u. 

u+i  l-u, 

1 . 

M  +  h 

u+l,  i; 

l-u,  u+i 

1; 

l-u, 

2~U,  2- 

1   „_Ll   1- 

u+i 

1   1. 

2>  2> 

i  i-u,  ^; 

^-u, 

1   1 

2')  2' 

i  u+l,  0; 

u+i 

i  0; 

i  l-u,  0; 

l-u, 

h   0. 

Sixteen  equivalent  positions: 

(k)  xyz;  yxz;  xyz;  yxz; 
yxz;  xyz;  yxz;  xyz; 
'l~x,  ^— y,  2~2:;    y+2>  2~x,  g  — z;    x+^,  y+2j  2"~z; 

2~y?  x+2,  2~z; 
¥-y,  ¥-x,  z+l;    |-x,  y-Hi  z+l;    y+|,  x+l,  z+|; 

X+2}     2~y}    2+2' 

Space-Group  D^. 


^0. 


(c)  OH;   loi 

(d)  iOO;    0^0. 


Two  equivalent  positions : 

(a)  0  0  0;     2 

(b)  2  ^  ¥  i 
Four  equivalent  positions: 

(e)  OOu;    OOu;    Hu;    Hu. 

(f)  Oiu;    0|u;    ^Ou;    ^  u. 

(g)  u,  u-l-i  0;    |-u;  u,  0;    u+i  u,  0; 
(h)  u,  u+i  i;    |-u,  u,  I;    u+i  u,  |; 

Eigf/if  equivalent  positions: 


u,  i—u,  0. 


(i)   uvO;    vuO;    v+i  u+|,  0;    u+|,  ^-v,  0; 

uvO;  vuO;  5— v,  |— u,  0;  i— u,  v+|,  0. 
(j)  uv^;    vu|;    v+i  u+i  |;    u+i  |-v,  |; 

uv|;  vu|;  |-v,  ^-u,  |;  ^-u,  v-Hi  |. 
(k)  u,  ^-u,  v;    u-f-i  u,  v;    ^-u,  u,  v;    u,  u+|,  v; 

vi,  \i-\-i,  v;    l-u,  ti,  v;    u+i  u,  v;    u,  |-u,  v. 

Sixteen  equivalent  positions: 

(1)   xyz;  yxz;  xyz;  yxz; 

y+ix-|-|,  z;  x-l-i,  i-y,  z;  ^-y,  i-x,  z;  |-x,  y+i,  z; 

xyz;  yxz;  xyz;  yxz; 

h-y,  |-x,  z;  i-x,  y-\-^,  z;  y-|-|,  x+h  z;  x+|,  |-y,  z. 

Space-Group  D^.* 

Two  equivalent  positions: 

<n^  000;    Hi  (b)  00^:    HO 


THE   TETRAGONAL   SPACE-GROUPS  D^-D^. 


93 


Space-Group  D^  (continued). 
Four  equivalent  positions: 

(c)  OJO;    iOO;    |0|;    OH- 

(d)OH;   hoh   OH;   iof. 

(e)  OOu;    OOu;    i  I,  u+|;    i  i  ^-u. 
EtgfTi^  equivalent  positions : 

(f)  O^u;  |0u;  i  0,  u+|;     0,  i  u+l; 
0|u;  ^Ou;  |,  0,  ^-u;     0,  §,  |— u. 

(g)  u,  u+i  i;  u+i  u,  i;  u,  |-u,  i;  |-u,  u,  |; 
u,  u+i  f;  u+l,  u,  I;  0,  |-u,  f;  ^-u,  u,  f. 

(h)  uvO;  vQO;  v+|,  u+i  ^;  u+i  §-v,  |; 
uvO;  vuO;  |-v,  ^-u,  |;  ^-u,  v+i  ^. 

Sixteen  equivalent  positions : 

(i)  xyz;  yxz;  xyz;  yxz; 
xyz;  yxz;  xyz;  yxz; 
l-x,  y+i  z+^;     ^-y,  |-x,  z+l;     x+|,  |-y,  z+^; 

y+i  x+l,  z+l; 
l-x,  y+i  ^-z;     |-y,  |-x,  f-z;     x+i  |-y,  |-z; 

y+2>   ^+2,    2~Z. 

Space-Group  D/h.* 

Ttyo  equivalent  positions: 
(a)  0  0  0;    HO. 
(b)OOi;    Hi 

Four  equivalent  positions : 
(d)iiO;    fiO;    ffO; 


(c)  0|u;    lOu. 


i  3  n 
4   4  U. 

111.        3.31.        131 

442>        44^>       44  ¥• 


(e)  Hi 

(f)  OOu;    OOu;    Hu;    Hu. 


^z'gr/if  equivalent  positions : 


(g)  uu  0;  uti  0 

uu  0;  u  uO 

(h)  uu|;  uu| 

uu|;  uu| 

(i)  uOv;  Ouv 

uOv;  Ouv 


u+i  Hu,  0;  u+i,  u+l,  0; 

Hu,  u+i  0;  Hu,  Hu,  0. 

u+i  Hu,  i  U+I,  u+i  I; 

5~U,  U+2,  2J  2~U,  2~U,  2- 

U+i  i  v;  i  u+i  v; 


Hu,  i  v; 


i-u,  V. 


(J)  u,  u+i  v;  u,  Hu,  v;  u,  Hu,  v;  u,  u+i  v; 
u+i  u,  v;  u+i  u,  v;  Hu,  u,  v;  Hu,  u,  v. 

Sixteen  equivalent  positions: 

(k)  xyz;  y+i  Hx,  z;  xyz;  Hy,  x+i  z; 

x+i  y+i  z;  yxz;  Hx,  Hy,  z;  yxz; 

xyz;  Hy,  Hx,  z;  xyz;  y+i  x+i  z; 

Hx,  y+i  z;  yxz;  x+i  Hy,  z;  yxz. 


94 


THE   TETRAGONAL   SPACE-GROUPS   D^-D^. 


Space-Group  D^.* 

Four  equivalent  positions : 

(a)  0  0  0;    HO;    H*;    0  0^ 
(b)OOi;    Hi;    OOf;    Hi 
(c)  Oiu;    iOu;    i  0,  |-u;    0, 

i  u-^|. 

Eight  equivalent  positions : 

(A\   XXX-       311.      111.      111. 
\p.)    444,       444;       i  4  4k )       444; 
313.        113.        133.        313 

4  4   4;       4   4   4;       4   4  4;       4   4  4- 

(e)  OOu;    Hu;    h,  h,  u+|; 
OOu;    Hu;    |,  h,  |-u; 

(f)  uuO;    uu|;    u+|,  ^-u,  0; 
uuO;    uu|;    |-u,  u+|,  0; 

0,  0,  u+§; 
0,  0,  l-u. 
u+l,  u-t-l, 
l-u,  |-u, 

Sixteen  equivalent  positions: 
(g)  xyz;    y+l,  |-x,  z; 
•   x-f-iy+i  l-z;    y,  X, 

X,  y,  Z+2;      2~Yf   2~X, 

|-x,  y+l,  z;    yxz; 
Space-Group  DiV* 


l-z; 


1. 

2  ; 

1 
2' 


xyz;    l-y,  x-|-|,  z; 

l_-u-      l_tr      1 

2 


X,  t-y;t-z;  y,  X,  f-z; 
X,  y,  z+l;  y+l,  x+i  z+l; 
x+i  |-y,  z;    yxz. 


According  to  the  previous  definitions  (page  33),  this  space  group  is  D, 


10 


and  the  following  one  is  D4h, 
NiggU's  descriptions. 

Two  equivalent  positions : 

(a)  0  0  0;    OOi 

(b)  H  0;    H  I- 

(c)  0|0;    |0|. 
Four  equivalent  positions: 


The  two  are  here  interchanged  to  conform  with 


(d) 
(e) 
(f) 


OH;   100. 


OOi; 
111. 

^  ^  4; 


00  i 

113 

ar  5  4- 


(g) 

(h) 

(i) 

(J) 

(k) 

(1)   uO| 

(m)  u  i  0 


OOu 
i  i  11 

2    2   U 

0|u 
uOO 
11  i  i 

U  2    2 


OOu; 
1 1  f, . 

2   2"; 

0|u; 
uOO; 

"2   2; 

OtiO; 

2^2, 


0,  0,  u-M; 

1 


2;    2;    '^r  2  ) 

h,  0,  I— u; 

Ou|; 

luO; 


u. 


0,  0, 

1 

2; 

0,  u+i 


1    1    1—11 

2;    2;    2        *^- 

1 
^; 


Ou^. 


u  w  2; 

u|0; 


OuO. 

ill    i 

2  U  2. 


Eight  equivalent  positions : 


(n)  uui 

uuf 
(o)  Ouv 

Ouv 
(p)  |uv 

|uv 
(q)  uvO 

tivO 


uu 


n  3. 

4; 


Qui;    uuf; 


uuj;    utif;    Qui 

u,  0,  v-l-l;    OQv;    u,.  0,  w+h 

u,  0,  |  — v;    OQv;    Q,  0,  I— v. 

u,  I,  v-l-l;    |Qv;    Q,  |,  v-|-|; 

u,  i  |-v;    |Qv;    u,  i  |-v 

vQ|;    uvO;    vu|; 

vQ|;    uvO;    vu|. 


THE   TETRAGONAL   SPACE-GROUPS   D^-D^i. 


95 


Space-Group  D^  (continued). 
Sixteen  equivalent  positions: 


(r)   xyz; 

y,  X,  z+l;    xyz;    y,  x, 

z+i; 

xyz; 

y,  X,  1- 

-z;    xyz;    y,  x, 

2     z; 

xyz; 

y,  X,  z+l;    xyz;    y,  x, 

z+l; 

xyz; 

y,  X,  1- 

-z;    xyz;    y,  x, 

Hz. 

Space-Group  DlS 

* 

According  to  the  previous  definitions  this 

group  is  D4b. 

Two  equivalent  positions: 

(a)  000 

0  0|. 

(c)  HI; 

HO. 

(b)OOi 

OOf. 

(d)Hi, 

Hi 

i^owr  equivalent  positions: 

(e)  0|0; 

|oi; 

100;    OH. 

(f)   OH, 

lOf; 

OH;   |0i 

(g)  OOu 

OOu; 

0,  0,  u+i;    0, 

0,  Hu. 

(h)Hu 

Hu; 

h  h  u+^;     i 

1     1      11 

2>     2~U. 

(i)    uui 

uuf; 

uuf;    uuf. 

(J)   uuf 

;   uu?; 

uuf;    uuf.' 

Eight  equivale 

nt  positions: 

(k)  0  A  u 

iOu; 

i  0,  u+l;    0, 

i  u-hl; 

0|u 

§0u; 

i  0,  l-u;    0, 

i  Hu. 

(1)    uOO, 

Ou^; 

uO|;    OuO; 

tiOO 

Ou|; 

uO|;    OuO. 

(m)  u  H 

|uO; 

u^O;    |u|; 

uH 

|uO; 

u^O;    |ui 

(n)  u  V  i 

vuf; 

uvf;    vuf; 

f 

uvf 

vui; 

uvf;    vuf. 

(o)  u  u  v 

;    uuv; 

u,  u,  v+l;    u, 

u,  v+i; 

ti  u  V 

;    utiv; 

u,  u,  l-v;    u, 

u,  Hv. 

Sixteen  equiva 

lent  positions: 

(p)  xyz; 

y 

X,  z+l;    xyz; 

y,  X,  z+l; 

X,  y, 

|-z;     yxz;     x,  y,     ^-z 

;   yxz; 

.   X,  y, 

z+^;    yxz;    x,  y,     z+| 

;   yxz; 

xyz; 

y 

,  X,  Hz;    xyz; 

y,  X,  Hz. 

Space-Group  Dli,* 

The  space-groups  D|J  and  D4h  are  interchanged  to  conform  with  the  descrip- 
tions of  Niggli. 

Four  equivalent  positions : 


(a)  0  0  0;    0  0^;    HO; 


(b)  OH;   HI; 

(c)  oof;    OOf; 

(d)0H;   Hf; 


HO; 

113. 
2  1  4> 

1  n  1  • 


Hi 

OH. 
Ill 

!■  2   4' 

n  1  3 
0^4. 


96 


THE   TETEAGONAL   SPACE-GROUPS 


^4h 


Space-Group  D^  (continued). 
Eight  equivalent  positions: 


(e) 
(f) 
(g) 
(h) 
(i) 
(J) 


1 1  n- 

4   4  ^J 

13   1. 

4   4  ?> 

111 

4   4   2) 

1  3  n. 

4    4  ^> 

OOu 

Hu; 

OOu 

Hu; 

0§u 

,    ^Ou; 

O^ti 

,    hOu; 

uOO 

Ou|; 

uOO 

,    Ou^; 

uH 

|uO; 

an 

|uO; 

UU  J 

;    uuf; 

tiu  1 

;    uuf; 

3  3  1. 

4  4   2) 


3  1  n-        3  1  1. 

4  4  '-'>        4   4   2  J 

3  1  n 

4  4  U. 

2>    U+2I 

1  l—ii- 

2>     2        ^> 

0,  u+l; 
0,  ^-u; 

^+2)     2) 

i  — 11      i- 

2  ">     2> 

0,  u+l,  0; 
0,  l-u,  0; 
u+l,  u+l,  f  ; 

1         n        1         11       3  . 
2— U,    I  — U,    4, 


0,  0,  u+i; 
0,  0,  l-u. 
0,  i  u+l; 
0,  i  ^-11. 
u+l,  I,  0; 
l-u,  i  0. 
u+i  0,  ^; 
l-u,  0,  i 

l-u,  u+i  i. 


Sixteen  equivalent  positions: 

(k)  xyz;    y,  x,  z+|;    xyz;    y,  x,  z+^; 

x+2>  y  I  2>  z;    y+2>  2~^)  2~z;     2~x,  2~y>  2j 

2~yj  X"r2)  2~z; 
i-x,  y+l,  z;    |-y,  |-x,  z-1-^;    x+|,  |-y,  z; 

y+i  x+i  z+l; 
xyz;    y,  x,  |-z;    xyz;    y,  x,  |-z. 
Space-Group  D]^.* 
This  group  is  DH  of  the  previous  definitions. 
Two  equivalent  positions: 

(a)  0  0  0;    Hi  (b)  0  0 1;    HO. 

Four  equivalent  positions: 


(c)  0^0;   OH;   hoh   |oo. 

(d)OH;  OH;  Hf;  ^oi 

(e)  Hi;   HI;   IH;   Hi 

(f)  IH;   iH;   HI;   Hi 

(g)  OOu;    OOu;    i  i  u+§;    i  i  Hu. 

igf/i^  equivalent  positions: 

(h)Oiu;    Hu;    i  0,  u-hi;    0,  i  u+|; 

OH;    ^Ou;    i  0,  Hu;    0,  i  f-u. 

(i)   uOO;    OuO;    i  u+i  §;    u-HH,  1; 

uOO;    OuO;    i  Hu,  I;    Hu,  h  h 

(j)   uO|;    OuA;    |,  u-hi  0;    u-h|,  i  0; 

uOi;    Ou|;    i  ^-u,  0;    |-u,  |,  0. 

(k)  u,  u-l-i  \;    u,  Hu,  i    u,  l-u,  i 

% 

u+i 

f; 

u+i  u,  i;    u-hi  u,  f;    Hu,  u,  i; 

^ 

-u,  u, 

i 

(1)  u,  u-fi  1;    u,  Hu,  i;    u,  l-u,  f; 

Q, 

u+i 

1. 

4> 

u+i  u,  f;    u-H,  u,  \;    Hu,  u,  f ; 

1- 

-u,  u, 

i 

(m)uuv;    u-l-i  Hu,  v-H;    uuv; 
uuv;    \i-\-\,  U-I-5,  Hv;    uuv; 


l-u,  u+l,  v-H; 
^-u,  Hu,  Hv. 


THE   TETRAGONAL   SPACE-GROUPS   Dlu-Dlh. 


97 


Space-Group  T)^  (continued). 
Sixteen  equivalent  positions : 

(n)  xyz;    y+i  |-x,  z+|;    xyz;    J-y,  x+|,  z+|; 
x+i  y+i  l-z;    yxz;    ^-x,  ^-y,  ^-z;    yxz; 
^-x,  y+i  z+J;    yxz;    x+i  ^-y,  z+|;    yxz; 
xyz;    |-y,  |-x,  |-z;    xyz;    y+i  x+|,  |-z. 
Space-Group  D]^.* 
Space-groups  D]^  and  D|^  also  are  interchanged, 
i^owr  equivalent  positions : 


(a)  0  0  0; 
(b)OOi; 
(c)  0|0; 

(d)OH; 


OOf; 

2  ^  2  > 
2   '-'  4  > 


1  1  n-    A 

2    2"?        2 

1   i  1- 
^   2    4> 

100; 

OH; 


ill 

2   2    2' 
113 
2    2    4- 

n  i  A 

t»  2    2- 

i  n  A 

2^4' 


Eight  equivalent  positions: 


(e)  OOu;    Hu; 


OOti; 
(f)   Oiu; 


1  1  f] . 

2  2   ^J 


iOu; 
|0u; 


1 

27 

1 

2> 

2> 

1 


2}    ^\2} 


l-u; 


u+l; 

1 


0,  0,  u+i; 

0,  0,  i-u. 

0,  i  u-h§; 

0,  h  i-u. 


0, 

0,  t-u; 

(g)  u,  u-l-i  J;    u-}-^,  u,  f;    u,  ^-u,  I; 

u,  u4-|,  f;    u-l-i  u,  i;    ti,  |-u,  f; 

(h)  uvO;    uvO;    |— u,  v-|-|,  0;    u+l, 


t-u,  u, 
i-u,  u, 
-V,  0; 


3  . 

4> 

1 

4- 


vu|;    vu§;    |-v,  |-u,  |;    v-fi  u-f-|,  |. 


Sixteen  equivalent  positions: 


(i)  xyz;  y,  x,  z+|;  xyz;  y,  x,  z-\-^; 
xyz;  y,  x,  §-z;  xyz;  y,  x,  |-z; 
l-x,  y+^,  z;    ^-y,  ^-x,  z+^;    x+i  |-y,  z; 

y-l-i  x-fl,  z-f-^; 
^-x,  y-Fi  z;     i-y,  |-x,  |-z;     x+i  |-y,  z; 

y   I    2">    X-pz,    2        2' 

Space-Group  D^.* 

Two  equivalent  positions : 


(a)  0  0  0; 


111 

2    2    2- 


(b)OOi;    HO. 


Four  equivalent  positions : 


(c) 
(d) 
(e) 
(f) 
(g) 


0|0 
n  A  i 

U  2  4 

OOu 
uuO 
uu| 


OH; 
n  1  3 . 

OOu; 
uu  0; 
uu^; 


hO^ 


^,    lOO. 
1  n  ^-  A  n  A 

2  "  4>   2  "  4- 
1   1   n_l_l- 

l-u,  u-f-i  ^; 
l-u,  u+i  0; 


1 

2> 


i-u. 


u+i  l-u, 

u+i  l-u, 


Ez'gi/ii  equivalent  positions : 


(h)O^u 

on 

(i)    uvO 

tivO 

(j)   uuv 

U  U  V 


|0u; 
*0u; 
vuO; 
vuO; 
uuv; 
uuv: 


I,  0,  u+h 
h  0,  l-u; 
v+l,  i-n,  I; 

i_v     ii4-A     A. 
2        V,     U-t-2,     2j 

u+i  §-u,  l-v; 
^-u,  u-M,  v-Hi; 


0,  h  u+l; 
0,  i  J-u. 
u+l,  l-v,  I; 
l-u,  v+i  i 
u+i  l-u,  v-H; 
|-u,  u+^,  l-\. 


98 


THE   TETRAGONAL   SPACE-GROUPS   Dli-D^t. 


Space-Group  D]^  (continued). 
Sixteen  equivalent  positions: 

(k)  xyz;    y+i,  |-x,  z+|;  xyz; 

xyz;    y+i  |-x,  |-z;  xyz; 

l-x,  y+i  z+l;  yxz;  x+i, 

|-x,  y-}-|,  |-z;  yxz;  x-HI, 

Space-Group  D1^.* 

Space-groups  D]^  and  Dj^  are  here  interchanged 

Two  equivalent  positions: 


i-y,  x+l,  z+l; 

y, 


1 

2 

1-y, 
i-y, 


x+l, 

z+l; 

2        ^> 


2  ~z; 
yxz; 
yxz. 


(b)  0  01; 


(a)  000;    Hi 
i^owr  equivalent  positions: 

(c)  OOu;    OOu;    h,  h  §-u;    h  h  u+i 

(d)  0|u;    |0u;    |,  0,  A-u;    0,  i  u+i 
^tgr/i<  equivalent  positions : 


(e) 

111 

4  4  4; 

3  13. 

4  4  4; 

3  3  1.    13  3. 

4  4  4;   4  4  4; 

3  3  3. 

4  4  4; 

13  1. 

4  4  4; 

113.   3  11 

4  4  4;   4  4  4- 

(f) 

uuO 

;  ti  u  0; 

u+l,  J-u,  1; 

l-u, 

u+l,  1; 

utiO 

;  u  u  0; 

u+i  u+l,  1; 

l-u, 

l-u,  i 

(g) 

Ouv 

;  Ouv; 

u+i  i  v+l; 

i-u, 

1,  v-f-l; 

uOv 

;  uOv; 

h   u+l,  l-v; 

i,  1- 

-u,  l-v. 

Sixteen  equivalent  positions : 

(h)  xyz;     y-f-^  |-x,  z+|;  xyz;     |-y,  x-|-|,  z+f; 

x+i  y+h  l-z;     yxz;  f-x,  ^-y,  |-z;    yxz; 

xyz;     |-y,  |-x,  z+|;  xyz;    y+|,  x-|-|,  z+|; 

|-x,  y-M,  i-z;     yxz;  x+|,  |-y,  |-z;     yxz. 

Space-Group  D1^* 


Four  equivalent  positions: 


(a)  000;    IH; 


001; 


(b) 

('p^  1  1  n-    111 

V.W     4   4   ^>        4   4    2 

fiO; 
|0u; 


113. 
5  2   4; 

3  11. 

4  4^; 


OOi; 
OOf; 
HO; 


f   2   "• 

111 
^2    4" 

13    1 

4    4^- 


111. 

4   4  7; 


(d) 

(e)  Oiu; 


A   3    1  . 
4   4  2; 


1  3  n 
4   4"' 

1      n      1  —  11- 
2,    U,     2        U, 


0,  h  u+i. 


E^'gf/ii  equivalent  positions : 


(f) 

OOu 

Hu; 

OOu 

Hu; 

(g) 

uuO 

uu|; 

uuO 

Qui; 

(h) 

uu| 

uuO; 

utii 

;  QuO; 

h  h  u+i;  0,  0,  u-f-i; 
i>  h  |— u;  0,  0,  |— u. 
u-M,  l-u,  I;    u+i  u+i  0; 

l-u,  l-u,  0. 

u+i  u-f-i  I; 


l-u,  u+l,  I; 
u+l,  l-u,  0; 
l-u,  u+l,  0; 


l-u,  l-u,  |. 


(i)   u,  u+l,  v; 
u+l,  u,  v; 


u, 


-u,  v+l;    u,  l-u,  v;    u,  u+|,  v+|; 


u+l   u,  |-v;    |-u,  u,  v:    |-u,  u, 


l-v. 


THE   TETRAGONAL  SPACE-GROUPS  DiS-DlJ. 


99 


Space-Group  D]^-{contimied). 
Sixteen  equivalent  positions: 

(j)   xyz;    y+l,  |-x,  z-l-|; 

"  '-hi 

X,  y,  z+l; 

l-x,  y+l, 


x-hi  y+i  z;    y,  x 


^-z; 


A— 7' 

2        ^j 


xyz;    l-y,  x-h|,  z+|; 
'-X,  l-y,  z;    y,  x, 
z;    X,  y,  z+l;    y+|,  x+^,  z; 
-z;    yxz;    x+i,  |-y,  |-z;    yxz; 


i-y, 


•J       X, 


Space-Group  DIJ. 


Two  equivalent  positions: 

(a)  0  0  0;    Hi 
Fowr  equivalent  positions : 

(c)  0^0;    ^0  0;    ^Of; 

(d)  OH;   |oi;   OH; 

(e)  OOu;    OOu; 
Eight  equivalent  positions: 


(b)  OOi;    HO. 


1    1 

2;     2) 


OH. 
*oi 

i  — IT        i 
2        l^;        2j 


U  +  |. 


(i)  4  4*; 

3  3    3. 

4  4    4; 

(g)  O^u; 

O^u; 
(h)  uuO; 

uti  0; 

uOO; 

tiOO; 


(i) 
(J) 


2  > 
1. 

2  ) 


13  1. 
4  4  4  7 
111. 

4    4    4> 

^Ou; 
|0u; 
uu  0; 
tiuO; 
OuO; 
OuO; 
Oui; 
Ou^; 


3  11. 

4  4  4> 
13  3. 
4   4   4> 


3.  1   1  . 
4    4    4; 
113 
4   4   4- 


2> 


h  0,  u+^; 
h>  0,  2~u; 

i  u+i  0 
h,  h-u,  0 


0,  h  u+l; 
0,  i  ^-u. 

1    ,,    1 „    1 

^— u,  ^— U,  5. 

u+i 


1     1 . 

2}    1> 


1 

2> 


u+l,  I,  0; 
^-u,  h  0. 


Sixteen  equivalent  positions: 


%  |-u,  i; 
"-U,  u,  I; 


0,  i 


•u,  f ; 


f 


■u,  ti,  i; 


(k)  u,  u-H,  i;    |-u,  u,  i; 

u+i  u,  f;    u,  ^-u,  f ;    § 

u+i  u,  f;    u,  u-l-i  f; 
u,  u+i  i;    u+l,  u,  i; 
(1)   uvO;    vuO;    uvO;    vtiO; 

vuO;    uvO;    vuO;    tivG; 

u+iv+l,  ^;    §-v,  u+il; 

v+iu+l,  i;    u+l,  l-v,  i; 
(m)uuv;    uuv;    uuv;    uuv; 

uuv;    uuv;    uuv;    uuv; 

u+i  l-u,  v+l;    u+i  u+i  v-Hl; 

Hu,  u+i  l-v;    u+i  u+i  2-v; 


u+i  u,  i; 
u,  u+l,  f ; 
^-u,  u,  f; 
u,  |-u,  h 


u,  I- 


1 
2 

1  —  v   i- 

2  V,    2 


V,  I;    v+iHu,  1; 
u,  I;    |-u,  v+i  |. 


|-u,  u+l,  v+^; 

|-u,  l-u,  v-l-l; 
u+i  l-u,  ^-v; 

l-u,  §-u,  i-v. 


(n)Ouv;    uOv;  Ouv;    uOv; 

uOv;    OQv;  uOv;    Ouv; 

i  u+iv-f-i;  l-u,  iv-l-l;    i^-u,  v-|-^;    u+i  i  v-l-^- 

u+l,  i  §-v;  i  Hu,  l-v;    ^-u,  i  |-v;    |,  u+i  -J-v. 


100 


THE   TETRAGONAL   SPACE-GROUPS   dII~J)1 


Space-Group  D^l-icontinued). 

Thirty-two  equivalent  positions: 

(o)  xyz;  yxz;  xyz;  yxz; 
yxz;  xyz;  yxz;  xyz; 
xyz;  yxz;  xyz;  yxz; 
yxz;  xyz;  yxz;  xyz; 
x+i,  y+f,  z+l;     l-y,  x-hl,  z+|;     ^-x,  |-y,  z+l; 

y+h,  l-x,  z+§; 
y+i  x+i,  ^-z;     x-f-|,  i-y,  |-z;     i-y,  |-x,  |-z; 

5~x,  y+2,  2~2; 
l-x,  l-y,  i-z;     y+l,  ^-x,  l-z;    x+^,  y+f,  §-z; 

i-y,  x+l,  ^-z; 
l-y,  ^-x,  z+l;     l-x,  y+l,  z+l;    y+l,  x-|-|,  z+|; 

x+i  l-y,  z+|. 
Space-Group  D^. 

Four  equivalent  positions: 


(a)  0  0  0; 
(b)OiO; 
(c)  OOi; 

(d)OH; 


001; 
100; 


111. 

2   2   2; 

n  1  1  • 

1  i   3  . 

2  2    4; 


2  vJ  4,        2  *-*  4> 


t  tw- 
in A 

2  "  2' 

4  i  i 

2    2   4' 

nil 

<J   2    4' 


Eight  equivalent  positions: 


/p^   1  in-     3  in-    iin-    iio- 

Vc^     4  4";       4  4^;       4   4^;       4  4  ^> 

(f) 


1   3   1. 
4  4  2; 


1   3.  1  • 
4   4  2^; 


OOu; 

OOu; 
(g)  0|u; 

0|u; 
(h)  u,  u+l,  i; 

u+i  u,  f; 


1  l^l- 
2    2    l-l; 

1  1  f,  • 

2  2"; 

AOu; 
|0u; 


111. 

4  4^; 

1 

1} 

1 

¥; 

1 

^; 

1 

^; 


3  11 

4  4   2' 


^;  ^+2; 

1 


2;     2       U; 

0,  u+l; 
0,  l-u; 


t-u,  u, 
u,  i-u, 


1 . 

4; 

1- 
4; 


0,  0,  u+l; 

0,  0,  i-u. 

0,  I,  u-f-l; 

0;  h  i-u. 
U;  |-u,  1;    u-M,  u, 
|-u,  u,  f;    u,  u-M, 


Sixteen  equivalent  positions: 


(i)  uuO;  uu| 

uuO;  uu| 

utiO;  uu| 

uu  0;  uu  I 

(j)  uOO;  u0| 

OuO;  Ou| 

uOO;  uO| 

OuO;  Oui 

(k)  uvi;  vuf 


vu 


uv| 


u  vi;  vti  f 


vui;  " 


u  vf 


1  1—11 

2;  2   " 


(1) 


u,  u+l,  v; 
u+i  u,  v; 
u,  -2--U,  i-v; 
^-u,  u,  v+l; 


u+l,  u+l,  0 
|-u,  u+l,  0 
l-u,  l-u,  0 
u+l,  |-u,  0 
u+l,  i  0 
i  u+i  0 
^-u,  I,  0; 

0; 

hi 

-I 

-7 

'V 

\ 

u-l-l,  u,  |-v; 
U;  u-Hi  v-f-f ; 


v+i 

u+i 

h 

u+i 

l-v, 

1. 

4; 

^-v, 

l-u, 

1- 

4; 

l-u. 

v+l, 

i; 

i-u, 

u,  v; 

u,  |- 

-u,  v; 

u-i-l,  u+h  h 

l-u,  u+l,  I; 

1  11   l_n   !• 

2  ^;  2   ";  2^» 

u+l,  i-u,  |. 

n-4-1  1  1- 
U-r2;  2;  2; 

1  n  _L1   1  • 
2;  U-j-2;  ^, 

1   11   1   1- 
¥  — U,  J,  2, 

I;  |  — U,  |. 

u+l,  v+l,  f; 

l-v,  u+l,  f; 

l-u,  l-v,  f ; 

v+i  l-u,  f. 
u,  l-u,  v;   u+l,  u,  v; 
l-u,  %  v;   u,  u+l,  v; 
u,  u+l,  l-v;  l-u,  u,  |-v; 
u+l,  u,  v+l;  u,  |-u,  v+|. 


THE   TETRAGONAL   SPACE-GROUPS   Dl^-bljj.' ,  :      ;  : "  lOJ: 

Space-Group  D^  (continued). 

Thirty-two  equivalent  positions: 

(m)xyz;        yxz;        xyz;  yxz; 

yxz;        xyz;        yxz;  xyz; 

X,  y,  i-z;    y,  x,  J-z;  x,  y,  ^-z;    y,  x,  |-z; 

y,  X,  z-\-i;    X,  y,  z+i;  y,  x,  z+|;    x,  y,  z+i; 

x+i  y+i  z+l;    |-y,  x+l,  z+i;     |-x,  ^-y,  z-f-^; 

y  I  2)  'a      X,   z-i-2> 
y+i  x+l,  |-z;    x+l,  ^-y,  |-z;     ^-y,  |-x,  ^-z; 

2     X,  y+2j  2     z; 
|-x,  |-y,  z;    y+i,  |-x,  z;    x+|,  y+i  z;    i-y,  x+|,  z; 
|-y,  l-x,  z;    ^-x,  y+l,  z;    y+i  x+i  z;    x+i  |-y,  z. 
Space-Group  T)]^* 

Four  equivalent  positions : 

(a)  000;    OH;    iOf;    Hi 

(b)  OOi;    OH;    ^01;    HO. 
Eight  equivalent  positions: 

lis.        313.       135.        3  f\  7 

5^48>458>^48>4'-'8' 
/JN  111.  inl.  131.  111. 
\yj    ^48;       4"8>       ^48;       428> 

ni5.     3.ni«    nil-    3.11 

^48>       4"8>       '-'48>       42   8' 

(e)  OOu;    0,  i  u+i;    i  i  u+|;    i  0,  u-hf ; 
OOu;    0,  i  i-u;    i  i,  Hu;    i  0,  |-u. 
Sixteen  equivalent  positions : 

(i)  U4  §;  4>  2~u,  g;        u  4  §;  T^f; 

n4_l     11.  3     i_,,     3.  i  —  n     3     5.      3,,  7. 

^^2>    4j     8>  4>    ?        "j     8>  2        l^)     4>     8>        4   *^  8) 

r,  11-  3      ,i_l_l      3.  nil.  3ril. 

l_n      15.  1      ii_Ll      3.  n_Ll      3      1.       1  f;   7 

^        ")     4)     8}  4>    U-t-2>     8j  U-t-2>     4)     8>        4  U  ||. 

(g)  uuO;  u,  Hu,  i;        uuO;  ^-u,  u,  f; 

u+i  u,  f ;         u-l-l,  Hu,  I;  Hu,  ti,  |;    u,  u+|,  i; 
uuO;  u,  i-u,  I;        uuO;  u,  u+i  |; 

2— U,    U-i-2>     2;        2~U,     2~U,     2;    U+2>    U,     4;       U+2>    U+2,     2- 

(h)  Ouv;  u,  i,  v+l;        Otiv;  Hu,  0,  v+f; 

i  u,  f-v;  u+f,  i  Hv;  I,  u,  |-v;    uOv; 

iu+iv+l;      u+i  0,  v+l;    i  Hu,  v-M;  u,  i  v+i; 

0,  u-M,  i-v;      u  0  v;  0,  i-u,  |-v;Hu,  |,  |- v. 

Thirty-two  equivalent  positions: 

(i)    xyz;    y,  Hx,  z-hi;  xyz;    |-y,  x,  z+|; 

X+-2-,  y,  f-z;    y+i  J-x,  §-z;    Hx,  y,  |-z;    yxz; 

xyz;    y,  |-x,  z+J;    xyz;    y+i  x,  z+f; 

§-x,  y,  |-z;    Hy.  Hx,  i-z;    x+|,  y,  f-z;    yxz; 

x+i  y+i  z+i  y+i  x,  z+f;  Hx,  Hy,  z+i 

y,  x+i  z+l; 
X,  y+i  i-z;  yxz;  x,  Hy,  l-z;  Hy,  x+i  Hz; 
Hx,  y+i  z+i-  Hy,  x,  z+f;  x+i  Hy,  z+i- 

y,  x+i  z+i; 
X,  y+i  i-z;  yxz;  x,  Hy,  J-z;  y+i  x+i  ^-z. 


•102 


THE   TETRAGONAL   SPACE-GROUP   D^. 


Space-Group  DlS-* 

Eight  equivalent  positions: 


(a)  000;    OH; 


uuf; 
(b)OOi; 


OH; 

0|0; 


100; 


1  1  n. 
111 

Hf; 
Hi 


Sixteen  equivalent  positions: 

2   4  5) 
'J  4   8> 


(c)  Oj  I;   i^  I; 


?  4   8 
7 

f 


Oi 

H 

(d)  0  0  u 
OOu 
Hu 
Hu 


(e) 


(f) 


11  i  i 

U  4   8 

uH 

uff 
uuO 

u4-^ 


3  15. 

4  ^  8> 

3  11. 

4  2   8; 

HI;   h 


3    3  . 

1  8> 


0,  h 

0,  i 
0,  i 
0,  i 


u+i; 
u+f; 

3_„. 
4        ", 


-u; 

2-u,  f; 
l-u,  I; 

U,    4>    8; 


4  '^  8> 

fOf; 
iOf. 

h  0,  u+i; 
i  0,  u-hf; 
i  0,  f-u; 
i  0,  i-u; 

f,    3    1  . 
U  4   8> 


1 

2; 
1 

2> 


uH; 

1 


2 


u; 
u: 


0,  0, 

0,  0,  u-f-i 


u,  t; 
u.  u-l-i  I; 


U~f"2j     4>    fj 

u,  l-u, 
u+i  ^-u,  0; 


1 

2  ~ 

3 

¥> 

1 

4; 
1  . 
4; 


1  . 
8> 
3  . 
8> 

1. 
8> 


uuO; 
-u,  u, 


4  U  8, 

3  „  1. 

4  U  8> 

4  U¥, 
1    f!    3 


?;  u-}-i  v 

uu^; 
Thirty-two  equivalent  positions: 


n   1- 
u>  4> 


u,  §-u, 


i; 

u, 

f; 


•u,  I; 


5-u,  u,  I; 
u  u  i; 
u,  u-f-l,  J; 
^-u,  u+^,  0. 


(g)  xyz;  y,  l-x,  z+i; 

x+i  y,  i-z;    yH-i,  ^-x,  z; 

X,  y,  z+i;    y,  l-x,  z+l; 

^-x,  y,  f-z;     ^-y,  |-x,  ^-z;    x-f-|,  y,  f-z;    yxz; 

x+i  y+i  z+l;    y+i,  X,  z+f;     ^-x,  |-y,  z+|; 


xyz;   i-y,  x,  z+f; 

i-x,  y,  i-z;    y,  x,  §-z; 
X,  y,  z+l;  y+l,  X,  z+i; 


y,  x+l,  z+J; 
X,  y+i  f-z;    y,  x,  |-z;    x,  |-y,  f-z;    |-y,  x+|,  z; 
^-x,  y+l,  z;    l-y,  X,  z+i;    x+|,  |-y,  z;    y,  x+|,  z+f; 
X,  y+l,  i-z;    yxz;    x,  |-y,  f-z;    y+|,  x+|,  ^-z. 


SPECIAL   CASES   OF  THE   CUBIC   SPACE-GROUPS. 


103 


CUBIC  SYSTEM. 
THE  SPECIAL  CASES  OF  THE  CUBIC  SPACE-GROUPS. 

ONE  Equivalent  Position. 
(la)  0  0  0.  (lb)  Hi 

TWO  Equivalent  Positions. 


(2a)  0  0  0; 

Hi 

THREE  Equivalent  Positions. 

(3a)  h  \  0; 

|0i; 

OH-                (3b)  10  0; 

OH; 

0  0|. 

FOUR  Equivalent  Positions. 

(4a)  uuu; 

uuu; 

tiuu;    utiu. 

(4b)  0  0  0; 

HO; 

|0|;    OH. 

(4c)  \\\, 

iOO; 

OH;   ooi 

(4d)Hi, 

HI; 

3  13.        331 

4  4   4;        4   4   4' 

(4e)  Iff; 

3  11. 

4  4  4; 

HI;   Hi 

(4f )   uuu 

;   u+i 

^-u,  u;    u,  u+i  ^-u; 

|-u,  u,  u+i 

(4g)  HI; 

IH; 

HI 

3    7    5 
8    8    8- 

(4h)H|; 

17    3. 
8    8    8  > 

III 

13    1 
8    8    8- 

(4i)  IH, 

HI; 

fH 

Iff. 

(4j)  Hi 

Iff; 

111 

8   8  5 

HI. 

SIX  Equivalent  Positions. 

(6a)  uOO 

;    OuO 

OOu;                (6e)  0^0; 

OOi; 

HO; 

uOO 

OuO 

ooti.                   HI; 

HO; 

OH. 

(6b)  iuO 

;    0|u 

uO|;              (6f)  OH; 

HI; 

HO; 

HO 

;   OH 

tiOi                   OH; 

fO^; 

HO. 

(6c)  Ou^ 

;   |0u 

uH;            (6g)HI; 

HO; 

OH; 

Ou| 

;   Hu 

uH.                   HI; 

fH; 

OH. 

(6d)|u| 

;   Hu 

uH; 

|Gi 

;   HQ 

QH 

. 

EIGHT  Equivalent  Positions. 

(8a)  uuu;  uuu;    u+|,  u+|,  u+|;    |-u,  u+|,  |-u; 

uuti;  uuu;    u+l,  |— u,  |— u;    |— u,  |— u,  u+l- 

(8b)  uuu;  |-u,  u,  ti;    u+|,  u+|,  u+|;    u,  u+|,  |-u; 

u,  u,  |-u;    u,  |-u,  u;    u+|,  |-u,  u;    |-u,  u,  u+|, 

(8c)  uuu;  uuti;    tiuu;    Qtiu; 

tititi;  uuu;    utiu;    uuti. 

(8d)  uuu;  tiuti; 

utiti;  titiu; 
(8e)  1 1 1; 


Hu,  l-u,  §-u;    u+l,  l-u,  u+l; 
|-u,  u+l,  u+l;    u+l,  u+l,  |-u. 


13   3. 

4  4  4; 

13.       1  1  1. 

4  4;       4  4  7; 


3  13. 

4  4   4; 


3  3  1. 

4  7  4; 


4   4*;       4  t  f • 


104 


SPECIAL   CASES   OF   THE   CUBIC   SPACE-GROUPS. 


EIGHT  Equivalent  Positions. — Continued. 


(8f)  OH; 

Hi; 

HO; 

0  0  0; 

13    3. 
4  4  4; 

3  13. 

4  4   4) 

3  3    1. 

4  4  4; 

iih 

(8g)  10  0; 

0|0; 

001; 

111. 

2  5^, 

3  11. 

4  4   4> 

13    1. 

4   4   4> 

HI; 

3  3    3 

4  4   4- 

(8h)  uuu; 

u+i 

l-u,  u 

;   ti,  u+- 

J;  Hu;    1 

-u,  u,  u+l; 

uuti; 

^-u, 

u+l,  u 

;    u,  l-u,  u+J;    u 

+i  u,  i-u. 

(8i)   0  0  0; 

HO; 

1  n  1  • 

OH; 

IH; 

OOi; 

OiO; 

|0  0. 

(8j)   uuu; 

u+i, 

l-u,  u 

;   u,  u+- 

h  i-u;    i 

-u,  u,  u+j; 

l-u, 

i-u,  i 

—  u;    u- 

■M,  l-u, 

uH;    f- 

-u,  u+l,  u+f; 
u+i  u+i  l-u 

(8k)  uuu; 

u+i 

l-u,  u 

;      U;    U  +  - 

11    ,,.    1 

2;    ^~U,       2 

-u,  u,  u+l; 

f-u, 

3_„       3 

4              ^)        4: 

-u;    u+i  i-u, 

u+f;    \- 

-u,  u+f,  u+f; 

u+f;  u+f,  f-u 

(81)  HI; 

HI; 

IH; 

III; 

Iff; 

IH; 

HI; 

III. 

(8m)  Iff; 

3    5    1. 

8   8   8; 

13    5. 

8   8   8; 

5    13. 

8   8   8; 

8    8    8; 

HI; 

5  11. 
8   8   8; 

17    5 
8    8    8- 

TWELVE  Equivalent  Positions. 

(12a)  u  0  0 

;    uOO 

u+l, 

hh   \- 

-u,  i  1; 

OuO 

;    OuO 

1      ,,_Ll      1.       1 
2;    ""r2;    5;       2; 

1    1,   1 . 

2— U,    ^, 

OOu 

;    OOti 

ii 

u+l;    i 

1;    l-U. 

(12b)  u^O 

;    u|0 

u+i 

0,  1;    ^ 

-u,  0,  1; 

Ou^ 

;    Ou^, 

1;  u+l,  0;    i 

l-u,  0; 

|0u 

;   iou, 

0;    i 

u+l;    0, 

1,  l-u. 

(12c)  uOi 

;   uH, 

u+l, 

13.       1 

^;    f;      2" 

-u,  0,  f ; 

iuO 

;   HI; 

f,  u4 

-I;i;      f; 

Hu,  0; 

Oiu 

;   HQ; 

1       3 

2;    4; 

u+l;   0, 

f.  l-u. 

(12d)  0  u  V 

;    Ouv 

;    Ouv; 

Ouv; 

vOu 

;    vOu 

;    vOu; 

vOu; 

uvO 

;    uvO 

;    uvO; 

uvO. 

(12e)  |uv 

;    5UV 

|uv; 

|uv; 

v^u 

;    v§u 

viu; 

v^ti; 

U  V2 

;  uv| 

uv|; 

u  v|. 

(12f)   uOi 

uOi; 

u|0; 

ti|0; 

|uO 

|uO; 

Ou|; 

Ou|; 

O^u 

0|u; 

Hu; 

|0u. 

(12g)  uuv 

;    uuv 

;    tiu V 

utiv: 

vuu 

;    vuu 

;    vuu 

vuu; 

u  vu 

;    uvu 

;    uvti 

uvu. 

(i2h)ni; 

|0f 

OH; 

OH; 

iiO; 

HO. 

HO; 

HO: 

OH; 

Ofi 

iO|; 

foi 

SPECIAL   CASES   OF   THE   CUBIC   SPACE-GROUPS. 


105 


TWELVE  Equivalent  Positions. — Continued. 


(12i) 

uOi 

,  uO^;  u+l, 

0,  h 

|-u,  0,  1; 

|uO 

;  ^uO;  i  u+ 

h   0; 

h   l-u,  0; 

0|u 

;  OH;  0,  i  u+^; 

0,  1,  l-u. 

(12j) 

u^O 

;  ti^;  u+l, 

h  0; 

l-u,  1,  0; 

Oui 

;  Oui;  0,  u+ 

1,1; 

0,  l-u,  1; 

^Ou 

;  \0u;    i  0,  u+i; 

1,  0,  l-u. 

(12k) 

fOi; 

iOf;  Hi; 

fH; 

ifO; 

HO;  HI; 

Hf; 

OH; 

OH;  HI; 

13  7. 

5  4  8, 

(121) 

13  3. 

2  4  8; 

HI;  OH; 

Ofl; 

3  13. 

8  2  4; 

Hi;  fof; 

f|0; 

. 

3  3  1. 

HI;  HO; 

|0f. 

4  8?; 

• 

(12m) 

uiiO 

;  uuO;  uuO; 

tiuO 

Ouu 

;  Ouu;  Otiu; 

Ouu 

uOu 

;  uOu;  uOu; 

uOu 

(12n) 

uu^ 

;  uu  ^;  ii  u  1; 

uu| 

|uu 

;  1  uu;  1  uti; 

|uu 

u^u 

;  u|u;  u|u; 

u|u 

(12o) 

u,  h- 

-u,  i;  u,  u+i 

f; 

u,  l-u,  f; 

U;  u+i  1; 

h  u, 

^-u;  f,  u,  u+l; 

f,  u,  l-u; 

f,  u,  u+l; 

l-u, 

i,  u;  u+i  i 

u; 

l-u,  f,  u; 

u+l,  f,  u. 

(12p) 

u,  h- 

-u,  f ;  u,  u+l, 

1 . 

4; 

U;  l-u,  f; 

U;  u+i,  f; 

f ,  u, 

|-u;  i,  u,  u+l; 

f,  u,  l-u; 

f,  u,  u+l; 

^-u, 

f,  u;  u+i  i, 

u; 

l-u,  f,  u; 

u+l,  f,  u. 

(12q) 

i-u, 

u,  1;  f-u,  1- 

-u,  1; 

u+f,  u+l,  f; 

u+f,  ti,  f ; 

i  i- 

-u,  u;  1,  f-u, 

l-u; 

i  u+f,  u+i; 

1;  u+i,  u; 

u,  i 

i-u;  l-u,  1, 

f-u; 

u+l,  f;  u+f; 

Q;  i  U  +  i. 

(12r) 

l-u, 

u,  f;  f-u,  1- 

-u,  1; 

u+f,  u+l,  1; 

U+f,  Q,  1; 

if- 

-u,  u;  1,  f-u, 

l-u; 

i  u+f,  u+l; 

1,  u+l,  u; 

11   3 

U;  ¥; 

f-u;  l-u,  f, 

f-u; 

u+l,  i  u+f; 

u,  1,  u+f. 

(128) 

|0i 

IH;  fOf; 

ill. 

4  8  2, 

iio, 

fH;  ffO: 

Hf; 

OH; 

HI;  ofi; 

HI. 

SIXTEEN  Equivalent  Positions. 


(16a)  uuu 

;  u+l. 

u+l,  u;  u+l. 

u, 

u+l; 

U; 

U  +  l, 

U+l; 

uuu 

;  u+l, 

^-u,  u;  u+l. 

% 

l-u; 

U; 

l-U; 

l-u; 

tiuu 

;  l-u. 

u+l,  u;  l-u. 

u. 

l-u; 

u, 

U  +  l; 

l-u; 

uuu 

;  l-u, 

|-u,  u;  |-u, 

u, 

u+l; 

u, 

l-u. 

u+|. 

(16b)  Hi; 

ill; 
HI; 
IH; 

SSI. 

8  8  8; 

HI; 

3  5  7. 
¥  8  ¥, 

5  15.    155. 

8  8  8;   888, 
5  7  3.   113. 
8  8  8;   8  8  8; 

¥  8  ¥;  ¥  ¥  s; 
fit;  1 1 1- 

106 


SPECIAL   CASES   OF   THE    CUBIC   SPACE-GROUPS. 


SIXTEEN  Equivalent  Positions. — Continued. 


(16c) 


(16d) 


(16e) 


13  7 

8    8  8 

15  1 

8  8  8 
111 

8   8  8 

15  7 

8    8  8 

uuu 
uuti 
uuu 
uuu 
uuu 
uuu 


111. 

8  8  8) 

ill. 

8  8  8  7 

1  1  1- 

8  8  8  > 

1  1  5.- 

8  8  8> 


111. 
8  8  8; 

1  1  1. 

8  8  8) 

111. 

8  8  8> 
111. 
8  8  8) 


3  5  3. 

8  8  8) 
5  5  5. 


8  8  8) 

111 

8  8  8- 


uuu; 
uuu; 
uuu; 
uuu; 


2  u,  2  U)  2  u; 
|-u,  u+i  u+l; 


u+i  l-u,  u+l; 


u+l; 


(16f) 


(16g) 


u+^,  u+l,  u+^; 
u+i  l-u,  |-u; 
|-u,  u+l,  ^-u: 

^-u,  |-u,  u+l;  u+l,  u+l,  |-u. 
u,  u,  ^-u;  ^-u,  u,  u;  u,  ^-u,  u; 
u,   u,  u+^;  u+l,  u,  u;  u,  u+^,  u; 
u+i  u+l,  u+l;  u+l,  ^-u,  u;  u,  u+|,  |-u; 

^-u,  u, 
u-i  u-|.  u-|;  ^-u,  u+l,  u;  u,  |-u,  u+|; 

U  +  5)  u,  |-u. 
uuu;  u,  u,  |  — u;  |  — u,  u,  Q;  u,  |  — u,  u; 
u+i,  u+i  u+i;  i-u,  u+i  f-u;  u+i,  f-u,  i-u; 

f-u,  i-u,  u+i; 
u+i  u+l,  u+l;  u+l,  |-u,  u;  u,  u+|,  |-u; 

l-u,  u,  u+l; 
u+f,  u+l,  u+f;  f-u,  u+f,  i-u;  u+f,  f-u,  f-u; 

i-u,  f-u,  u+f. 
uuu;  u,  u,  |— u;  |— u,  u,  u;  u,   |  — u,  u; 
i-u,   i-u,  i-u;    u+l,  l-u,   u+f;  f-u,  u+f,  u+f; 

u+f,  u+f,  f-u; 
u+l,  u+l,  u+l;  u+l,  |-u,  u;    u,   u+|,  |-u; 

l-u,  u,  u+l; 
f-u,  f-u,  f-u;  u+f,  f-u,  u+f;  f-u,  u+f,  u+f; 

u+f,  u+f,  f-u. 


(16h) 


(16i) 


000; 

l|0; 
1 . 


10^ 


"22, 

1  1  1- 

8  8  8) 

7  7  7. 

8  8  8) 
3  3  3. 
8  8  8) 
5  5  5. 
8  8  8) 


1  1  1- 

4  4  4) 

1  1  1- 

4  4  4) 

113. 
4  4  4, 

111. 

4  4  4, 

1  1  1- 
8  8  8, 
115. 
8  8  8) 
111. 
8  8  8) 

1  1  1- 
8  8  8) 


3  1  3. 

4  4  4, 

1  1  1- 

4  4  4) 

1  1  !• 
4  4  4) 

111. 

4  4  4) 

1  1  1- 
8  8  8) 
111. 
8  8  8) 
3  5  1. 
8  8  8) 
113. 
8  8  8, 


2^2) 

001; 
0|0; 

10  0. 

111. 

8  8  8) 
111. 
8  8  8> 
5  13. 

8^8, 
111 
8  8  8* 


TWENTY-FOUR  Equivalent  Positions. 


(24a) 


uOO 

aoo 

OuO 
OtiO 
GOu 
GOu 


u+l,  I,  0;  u+l,  0,  I;  u||; 
2~u,  2)  0;  2~u,  0,  2)  ^2^^; 
I,  u+l,  0;  |u|;  0,  u+|,  |; 
I,  l-u,  0;  |u|;  0,  |-u,  |; 
||u;  I,  0,  u+l;  0,  |,  u+|; 
l|u:    I,  0,  l-u;    0,  I,  l-u. 


SPECIAL   CASES   OF   THE    CUBIC   SPACE-GROUPS. 


107 


TWENTY-FOUR  Equivalent  Positions. — Continued. 


(24b) 


(24c) 


(24d) 


(24e) 


(24f) 


(24g) 


(24h) 


i  ill 

4  4  U 

3  3  „ 

4  4  U 

11  i  i 
U  4  4 

Oi  i 

U  4  4 

i  n  i 

4  U  4 

13  1 
?  4  4 

4  ?  2 

^  i  4 

4  4? 

Ouv 

Otiv 

Ouv 

Ouv 

vOu 

vOu 

uOi 

iuO 

Oiu 

uOf 

f  tiO 

Ofu; 

uO| 

^uO 

0|u 

uO| 

no 
on 

UU  V 

V  uu 
u  vu 
uu  V 
vuu 
u  vu 

ioo 

Oi 
00 
fO 


0 
1 

4 

0 
Of  0 

oof 


r,  3  1 

u  4  4 

„  1  3 

U  4  4 

4  U  4 

3  ,-,  1 

4  U  4 

1  Tl  a 

4  U  4 

3  „  3 

4  U  4 

1  3  n 

nil 

U  4  4 
3 

1 
2 

1 

4 
3 
4 


01 

3  3 

4  4 

H 

3  1 

4  ? 

vOu 
vOu 
uvO 
uvO 
uvO 
uvO 

aH 

uu 

11  1  3 
U^  4 

3  n  1 
jU  2 

?  4  U 

u|0 
Ou^ 
iOu 
u§0 
Qui 
|0u 
ti  u  V 
vuu 
u  vu 
uti  V 
vuu 
u  vti 

13    1 

2    4? 

1  i  i 

4  2? 

i  i  1 
2    2    4 

111 
?  4  2 
111 

T  ?  ? 


3. 

4) 
3 

4; 
1 

4; 


?-u, 
?-u, 

„_l_l       1       3. 
U-ri,     4)     4> 

3     n_L.i      1. 
4>    U-t-2>     4> 

11        n       !• 
4 )     ?        ".'     4  ; 

3  1  n.      3  3  rj. 

4  4   U,        I  4  "> 


4> 
3 

4> 
1 
4> 
1      3 
4;    ¥> 


nil 

U  4   4 

1  n  1 

4  U  4 

111 

2  4  7 

111 
4  2  4 
111 
4   4? 

1 

2> 

1 

?> 

1 

?> 

1 

?> 


0  1  1- 

"  4    4> 

1  n  1. 

4  ^  4> 
113. 

111. 
4  ?  4> 

Hi 

u+i  v+i 
l-u,  ^-v 
u+i  ^-v 
l-u,  v+l 

v+?,  ?,  u+l 

1        17-       1       1        n 

^-u,  0.  f; 
I  l-u,  0; 
0,  i  i-u; 


1—11    1- 

2         ll>     4> 

u+i  f; 

?-u; 
?-u; 


u+i  0,  i; 

i  u+i  0; 

0,  h  n+h 

u+i  h  0; 

0,  u+i  ^; 

i  0,  u+l; 

§— u,  i  0; 

0,  ?  — u,  ^; 

h,  0,  ^-u; 

u+i  u+i  v+i; 

v4-?,  u+i  u-l-^; 

u+i  v+i  u+i; 

u+?,  ?-u,  |-v; 

l-v,  u+i  |-u; 

l-u,  l-v,  u+i; 


1        IT      1       nJ-l- 

v4-l     1     1  —  11- 

VT^2>     2»     2        U, 

u+?,  v+i  I; 
l-u,  i-v,  ^; 

?-u,  v+i  i 
u+?,  ?, 

h  i  u+i 
?-u,  i  ii 

1     i_n 

4>     2        U, 

?,  i  ^-u. 
u+i  0,  I; 
i  u+i  0; 
0,  h  u+i 


l-u,  0,  I 
i  ^-u,  0; 
0,  h  h-u. 

|-u,  u+l,  ^-v; 

l-v,  l-u,  u+l; 

u+?,  ?-v,  ^-u; 

^-u,  l-u,  v+l; 

v+?,  ?-u,  l-u; 

^-u,  v+i  l-u. 


Hi; 


3  1  n- 

4  ?^; 

13  0- 
?  4  U, 

HO; 
HO; 

OH; 

\0h 


n  3  1 . 
u  4  ?> 

3  n  1  • 

4  'J?; 

|0i; 

OH; 
OH. 


108 


SPECIAL  CASES  OF  THE   CUBIC  SPACE-GROUPS. 


TWENTY-FOUR  Equivalent  Positions. — Continued. 


(24i) 


(24j) 


(24k) 


(241) 


(24m) 


uOi; 

^-u,  0,  f 

;   uH; 

i  f-u,  0; 

HO; 

f,  l-u,  0 

4  ^  2  J 

l-u,  0.  i; 

Oiu; 

0,  I  i-n 

1  1  ,1  • 

2  4   U, 

0,  h  l-u; 

i  u+i  I 

f,  i-u,  1 

,,_1_1       1       3. 

i  u+l,  0, 

u+i  1   i 

i-u,  i  1 

i  u+l,  1; 

u+f,  0,  f ; 

i  i,  u+l 

i  h  i-u, 

1,  i  u+§; 

0,  1,  u+|. 

uuO;    uu 

0;    u+l,  1- 

-u,  1;    ^-u, 

1     „    1 . 

2— U,    ^, 

Outi;    Ouu;    ^,  u+l, 

l-u;    i  ^- 

-u,  i-u; 

liOu;    u  0 

u;    5-u,  i 

u+l;    l-u, 

11     „. 

2)    ^  — U, 

uuO;    uu 

0;    u+l,  u+l,  ^;    ^-u, 

u+i  f; 

Ouu;    Ouu;    ^,  u+^, 

u+i;    i  i- 

-u,  u+l; 

uOu;    uO 

u;   u+i  h 

u+^-;    u+l, 

i  l-u. 

u,  l-u,  i 

;    u,  u+i  f 

;    u,  ^-u,  f 

;    u,  u+l,  f; 

i  u,  ^-u 

,    h  u,  u+f 

;    i  u,  ^-u 

;    i  u,  u+l; 

5-u,  i,  u 

u+i  h  u 

;    ^-u,  f,  u 

;    u+l,  1,  u; 

u+i  u,  f 

2-u,  ti,  i 

;    u+l,  u,  1 

;    l-u.  u,  f; 

u,  f,  u+i 

ti,  i  l-u 

;   u,  i  u+^ 

;    u,  f,  l-u; 

i  u+i  u 

i  ^-u,  ti 

,    i,  u+i  u 

;    1,  l-u.  u. 

uOi; 

U+2>    ?>    4) 

l-u,  0,  1; 

QH; 

iuO; 

i  u+l,  1; 

f,  ^-u,  0; 

iai; 

0|u; 

i  i  u+l; 

0,  f,  i-u; 

HG; 

h  i-u,  0; 

3       3        1,       1  . 
4>     4~U,     2j 

h  u+f,  i; 

i  u+i  0; 

i-u,  0,  I; 

f-u,  i  1, 

u+i  i  i, 

u+i  0,  f ; 

0,  i  i-u; 

i  i  l-u; 

i  i  u+f; 

0,  i  u+i 

u,  u+i,  1 

u,  i-u,  1 

;   u,  l-u,  1 

;    u,  u+i  f ; 

i  u,  u+i 

i  u,  i-u 

;    i  u,  l-u 

;    f,  u,  u+l; 

u+i  i  u 

i-u,  i  u 

;   f-u,  i  u 

;    u+i  i  ti; 

u+i  i-u 

,  1;   i-u,  u 

+h  h  u+^ 

\,  u+i  f; 

l-u. 

i  u+i  i- 

-u;    i  l-u 

,  u+J;    1,  u 

+1,  u+l; 

i  l-u,  l-u; 
i-\i,  I,  u+l;  u+i  I,  l-u;  u+i  i  u+|; 

|— u,  f,  |— u. 
(24n)  u,  i-u,  I;  ti,  u+i  |;  u,  u+i  f ; 
i  u,  u+l;  i  u,  u+l; 
u+i  i  u;  u+i  i  u; 
u+l,  u+i  I;  l-u,  l-u,  I;  u+|,  |-u,  f; 

l-u,  u+i  I; 
u;  i  u+l,  f-u; 


u. 

l-u,  i; 

i 

u,  l-u; 

1- 

-u,  i  u; 

ti,  l-u,  f; 
i  ti,  l-u; 
l-u,  i  u; 


I,  u+l,  u+l;  i  l-u,  i 
u+i  I,  u+l;  l-u,  i  I- 


i  l-u,  u+l; 
l-u,  f,  u+l; 

u+i  f,  |-u. 


SPECIAL   CASES   OF  THE   CUBIC   SPACE-GROUPS. 


109 


TWENTY-FOUR  Equivalent  Positions. — Continued. 

(24o)  Ouv;  Ouv 

vOu;  V  0  u 

u  vO;  u  V  0 

uOv;  uOv 

Ovu;  Ovu 

vuO;  vu  0 
(24p)  |uv; 


u|v; 


|u  V 

uv| 
u^  V 
h  vu 


vuf;  ?u^ 

(24q)  uu v;  uu v 

vuu;  vuu 

u  vu;  ti  vu 

tiuv;  uuv 

vuti;  vuu 

tivu;  uvti 

(24r)  Ouv;  Ouv 

vOu;  vOu 

uvO;  tivO 

h  ^-v,  ^-u 

(24s)  u,  ^-u,  I 
h  u,  |~u 
i-u,  i,  u 
u,  u+i  f 
i  u,  u-f-^ 
u+i  f,  u 

(24t)  u,  l-u,  i 
i  u,  ^-u 
^-u,  i  u 


Ouv 
vOu 
uvO 
uOv 
Ovu 
vuO 
iuv 
v^  u 
uv^ 
u|  V 
I  vu 
vu^ 
uuv 
vuu 
u  vu 
uuv 
vuu 
u  vu 
Ouv 
vOu 
uvO 


Ouv 
vOu 
uvO 
uOv 
Ovu 
vuO 
|u  V 
v^u 
ti  v^ 

U§  V 

^  vu 
vu^ 
uti  V 
vuu 
uvti 
uuv 
vuu 

U  V  u 

Ouv 
vOii 
iivO 


l-u,  u,  I 
h  l-u,  u 


h,  v+i  u+^ 
v+i  u+i  ^ 

i  u,  u+^ 
u+i  i  u 
u,  l-u,  i 
i  ti,  ^-u 
i-u,  i,  ti 
u,  u+i  f 
i  u,  u+l 
u+i  i  u 
u+l,  u,  I 
u,  I,  u+l 
i  u+l,  u 


l-u,  i  v+^;  u+l,  ^,  l-v; 
i  v+i  |-u;  i  l-v,  u+^; 
v+il-u,  I;    i-v,  u+i  |. 


ti,  ^-u,  f 
i  ti,  ^-u 
^-u,  i  u 
u,  u+i  i 
i,  u,  u+l 
u+i  i,  u 
ti,  i-u,  f 
i  ti,  ^-u 
f,  ti 


f,  1    1 

11,  4,    2 

1  1. 

4,  2 


ti,  u+i  i; 
i  u,  u+i; 
u+i  i  a; 
u,  l-u,  f; 
i  u,  l-u; 
i-u,  i  u. 
ti,  u+i  i; 
h  ti,  u+l; 
u+i  i  ti; 
u+i  u,  i; 
u,  i  u+§; 
i,  u+i  ti. 


^-u,  u,  f; 
■u; 
■u,  ti; 
(24u)  uuv;    uuv;    tiuv;    titiv; 
vuu;    vuu;    vtiu;    vuti; 
uvu;    tivu;    uvti;    tivQ; 
i-u,  ^-u,  §-v;    l-u,  u+l,  v+l;    u+l,  |-u,  v+J; 

u+i,  u+^,  l-v; 
^-v,  l-u,  J-u    v+i  ^-u,  u+^;    v+i  u+^,  ^-u; 

^-v,  u+i  u+^; 
•u,  l-v,  ^-u;    u+i  v+l,  ^-u;    ^-u,  v+|,  u+^; 

u+i  l-v,  u+i. 


^-^ 


no 


SPECIAL   CASES  OF  THE   CUBIC   SPACE-GROUPS. 


TWENTY-FOUR  Equivalent  Positions. — Continued. 


(24v) 

lOi; 

HI 

,    fO|; 

7  11. 
8^4; 

i|0; 

H^ 

HO; 

iH; 

OH; 

HI 

>       ^  4   8> 

117. 
^4   8, 

|0|; 

IH 

113. 
>        8   2    4, 

fOi; 

f|0; 

HI 

,   HI; 

ifO; 

OH; 

113 

?  4   8 

IH; 

OH. 

(24w) 

fOi; 

113 
8  4  4 

iOf; 

5  11. 
8^4, 

HO; 

3  7    1 

4  8^ 

HO; 

15    1. 

4   8  5, 

OH; 

HI 

OH; 

115. 
5  ¥   8, 

fOl; 

Hi 

3    13. 

8^4; 

|0i; 

HO; 

HI 

4   8  ^> 

HO; 

OH; 

Hi 

13    3. 

1     4:     8> 

OH. 

THIRTY-TWO  Equivalent  Positions. 


(32a)  u  u  u 
uuti 
tiuu 
utiu 
uuu 
uuu 
utiu 
uuu 

(32b)  uuu 
utiu 
tiuti 
tiuu 

i-u,  i 


u+l 
u+l 
l-u 
l-u 
l-u 
l-u 
u+l 
u+l 
u+l 
u+l 
l-u 
l-u 


1 

2 

l-u,  u;  I 


u+l,  u;  u+l,  u,  u+l 

|-u,  u;  u+l,  u,  |-u 

u+l,  ti;  |-u,  u,  |-u 

u,  u;  |-u,  u,  u+l 
u,  u,  |-u 

u+l,  u;  |-u,  u,  u+l 

|-u,  u;  u+l,  u,  u+l 

u+l,  ti;  u+l,  u,  |-u 

u+l,  u;  u+l,  u,  u+l 

|-u,  u;  u+l,  ti,  |-u 

u+l,  ti;  |-u,  u,  |-u 

|— u,  u;  |-u,  ti,  u+l 
-u;  l-u,  f-u,  l-u; 


u. 

u+l, 

u+l; 

u. 

l-u. 

l-u; 

u, 

u+l. 

l-u; 

u. 

l-u. 

u+l; 

u, 

l-u. 

l-u; 

u, 

u+l, 

u+l; 

u, 

l-u. 

u+l; 

u. 

u+l. 

l-u. 

u, 

u+l. 

u+l; 

u. 

l-u. 

l-u; 

Q, 

u+l. 

l-u; 

u. 

l-u. 

u+l; 

t-u,  t-u,  f-u; 

4  — u,  4— u,  4— u; 
i-u,  u+i,  u+i;  f-u,  u+f;  u+i;  f-u,  u+i,  u+f ; 

l-u,  u+f,  u+f; 
u+l,  l-u,  u+l;  u+f,  f-u,  u+l;  u+f,  |-u,  u+f; 

u+l,  f-u,  u+f; 
u+l,  u+l,  l-u;  u+f,  u+f,  l-u;  u+f,  u+|,  f-u; 

u+l,  u+f,  f-u. 
(32c)  uuu;  u+l,  u+|,  u;  u+|,  u,  u+|;  u,  u+|,  u+|; 
uuu;  u+l,  |-u,  ii;  u+|,  u,  |-u;  u,  |-u,  |-u; 
uuti;  |-u,  u+l,  u;  |-u,  u,  |-u;  ti,  u+|,  |-u; 
uuu;  |-u,  |-u,  u;  |-u,  u,  u+|;  u,  |-u,  u+|; 
u+l,  u+l,  u+l;  u,  u,  u+l;  u,  u+|,  u;  u+|,  u,  u; 
|-u,  u+l,  l-u;  u,  u,  l-u;  u,  u+|,  ti;  |-u,  u,  u; 
u+l,  l-u,  l-u;  u,  ti,  |-u;  u,  |-u,  u;  u+|,  u,  u; 


|-u,  |-u,  u+l;  ti,  ti,  u+l; 


u,  l-u,  u;  I 


u,  u,  u. 


SPECIAL   CASES   OF   THE    CUBIC   SPACE-GROUPS. 


Ill 


THIRTY-TWO  Equivalent  Positions. — Continued. 

(32d)  i  I  i 


(32e)  If 


8  8  ) 

1  3. 

8  S} 

1  1- 

8  8  ) 

3  3.. 

8  8  > 

5  7. 

8  8  ) 

3.  1- 

8  8  } 

1  5  . 

8  8  7 

1  1  • 

8  8> 

3  3. 

8  8> 

3  1. 

8  8  7 

5  5. 

8  8  > 

1  5. 

8  8  ; 

3  5. 

8  8  f 

1  5. 

8  8> 

5  3. 

8  8  ; 

5  7. 

8  8  7 


17  7 

8  8  8 
113 

8  8  8 

7  5  7 

8  8  8 
3  3  5 
8  8  8 

7  5  3 

8  8  8 

1  k  1 
8  8  8 
3.  1  5 
8  8  8 
111 
8  8  8 

111 
8  8  8 

111 
8  8  8 
5  7  5 
8  8  8 
115 
8  8  8 

7  5  5 

8  8  8 
115 
8  8  8 
5  5  1 
8  8  8 
111 
8  8  8 


(32f)  uuu;  u,  u, 

4    »J>  4 


uuu; 

u+i 


111 

8  8  8 

111 

8  8  8 

17  5 

8  8  8 

5  5  i 

8  8  8 

5  1  1 

8  8  8 

111 
8  8  8 
13  5 
8  8  8 
15  5 
8  8  8 

111 
8  8  8 
3  7  7 
8  ?  8 

111 
8  8  8 
15  1 
8  8  8 

111 

8  8  8 

5  1  1 
8  8  8 

111 

8  8  8 

5  1  1 
8  8  8 

l-u; 
-u; 


111 

8  8  8 
15  5 
8  8  8 
113 
8  8  ¥ 
5  3.  7 
8  8  ¥ 
111 
8  8  8 
111 
8  8  8 
111 
8  8  8 
111 
8  8  8 

111 
8  8  8 
113, 
8  8  8 

111 

8  8  8 

111 

8  8  8 

111 
8  8  8 
111 
8  8  8 

111 
8  8  8 
5  11 
8  8  8 


u,  u,  u-hl; 

u+i,  u+i; 


u+i  u+^,  U+-2-; 


1 

4 

-u, 

1 

4 

-u, 

3 
4 

-u; 

1 
2 

-u, 

1 

2 

-u, 

1 

-u; 

u 

+i 

u 

+i 

u 

_i_i. 

T^4> 

u,  u,  u;  u,  t  — u,  u; 
u+i  i-u,  u+f;  i-u,  u+f,  u+i; 
u+f,  u+i  i-u; 
u-f-l,  ti,  u;  u,  u+i  u; 
i-u,  u+i  f-u;  u+i  f-u,  i-u; 

f-u,  l-u,  u+l; 
u+l,  |-u,  u;  u,  u4-i  ^-u; 

l-u,  u,  u+l; 
u+f,  f-u,  u+i;  f-u,  u+i,  u+f; 

u+i  u+f,  f-u; 
|-u,  u+l,  u;  u,  |-u,  u+^; 

u+l,  u,  |-u; 
f-u,  u+f,  i-u;  u+f,  i-u,  f-u; 
i-u,  f-u,  u+f. 


FORTY-EIGHT  Equivalent  Positions. 


(48a)  1 1  u 


iiu 


1  3 

4  4 

3  1  f, 

I  lu 

4  4  U 

II  i  i 
U  4  4 

fl  i  ^ 

U  4  4 

fl  ^  i 

U  4  4 

n  1  1 

U  4  4 

i  11  i 

4  U  4 

1  f,  1 

4  U  4 

1  y,  3 

4  U  4 

3  „  1 

4  U  4 


3 
4 

1 

4 

u 

1 
4 

3 
4 

u 

1 
4 

1 
4 

ti 

u 

1 
4 

1 

4 

u 

3 

4 

1 
4 

u 

1 

4 

3 
4 

fl  i  i 

U  4  4 

1  f,  1 

4   U  4 

1  1,  1 

4    U  4 

1  1,  1 

4   U  4 

i  fl  1 

4  U  4 


,,_|-1      1  i- 

l^    I    2,     4,  4> 

1  —  ll       1  1- 

2  Uj     4j  4> 

l_n     1  1- 

2         LI,     4,  4> 

„J_1      1  1. 

l-l    I    2j     4,  4> 

1      ii-l-l  1- 

4,     ^^    I    2j  4, 

1    l_n  1- 

4,     2         ^,4, 

1      1_,]  1- 

4  >     2         <-l>  4  » 

1  u_l_l  1- 
4,     Ui-2,  4, 

1_11       1  1- 

2  ",     4,  4, 

,l_Ll      1  1. 

UT^2,     4,  47 

n_|_l      1  1- 

UT^27     4»  4> 

l_il      1  1- 

2         lA;     4  7  4  > 


1  l  —  i]       1- 

47  2         1^7     4  7 

f7  U  +  l, 

h  u+l. 


47 
1. 

47 

i— 11    i- 

2        U,     4, 


u+^; 
^-u; 
i-u; 
u+^; 
5-u; 

1     1    n-i-l- 

47     47     ^^2f 

1      3      ,i4.1. 
47     47     l^    I    27 

3       1       1        11 
47     47     5  — U. 


1  1 

47  47 

1  1 

47  47 

i  1 

47  47 

1  1 

47  47 

1  1 

47  47 


112 


SPECIAL   CASES   OF  THE   CUBIC   SPACE-GROUPS. 


FORTY-EIGHT  Equivalent  Positions.— Continued. 


(48b) 


(48c) 


Ouv 
Ouv 
Ouv 
Ouv 
vOu 
vOu 
vOu 
vOu 
uvO 
uvO 
uvO 
uvO 
uOO 
uOO 
OuO 
OuO 
OOu 
OOu 
l-u 

1 
i) 

1 


i-u, 


v; 
,  v; 
i  u+l,  v; 
h  2-u,  v; 

^-v,  I,  u; 

v+i  i  u; 
u+i  v+i  0 
^-u,  |-v,  0 
u-Hl,  i-v,  0 
i-u,  v+i  0 
u+l,  i  0; 
l-u,  i  0; 
h  u+i  0; 
i  l-u,  0; 

11,-1. 

2    2  U, 
i. 

1       1  • 

4,)     if 

1. 

4> 


i  u,  v+l; 
1    f,    1  _ 

h  u,  l-v; 


v;    0 
0 

i  u,  v-f-l;    0 
v+i,  0,  u+i 


0,  u-hl,  v-l-^; 
0,  i— u.  ^— v: 


■2> 

l-v,  0,  ^-u 
i-v,  0,  u-Hi 
v+h  0,  i-u 

u+l,  V,  i 

l-u,  V,  § 

u+i  V,  ^ 

l-u,  V,  ^ 

u+i  0,  I 

u,  0,  I 


1 

2 

1  n    1 

2  U^ 

1  ii   1  • 


u+i  i-v; 


V,  i  i-u; 
u,  v+i  i; 


u,  h 


V,  i; 


ti,  v-|-§, 
It 


u|i 


Q^ 


4) 


f-u, 


3       1 


i> 


h  0,  u-f-l; 

3_„      1 

4        "^J     4> 

3 


0, 
0, 
0, 
0, 


2        '^J     2  ) 


3  . 

4> 


3      3.. 

7    4;    4> 


-U 


1     ii_|_i      1- 
4>     "14)    4> 


ii_l_3       3.      1.        ■n_l_3.      1  3.        n_I_l       3       3. 

U-r4>     4>     4)       U-t-4,     4,  4,       U-t-4,     4,     4, 

1      3_,,       1.        3       l_,i  3.        '        - 

4>     4        ^i     if       4)     4        "^j  4j 

3      „  J^3       1  .        3      ,,_ll  3  . 
"    I    4j 


3      n_|_3       1 
4>    U-^4,     4, 


3 

4j 

3       1 
4)     4> 


3 
4» 

4        U, 


1  3_,,       3. 

4j  4        *^i     4j 

1  ii_]_3       3.. 

J)  ^"Ti,    4) 

1  3       3_,,. 

4j  4»     4        "> 


(48d) 


4j     4)    4        U>        4)     4>    4        U>        4j     4)    4        ^j        4j     4»     4        ^J 
1      1      -ii-Ll-        3       3      ,,_Ll.        3       1      ,,_J_3..        1       3      -,    I    3 

4>   4>  U-hj,     4,  1,  U-I-4,     4,  t,  U-f-f,     4,  :j,  Ui-:f. 

uuv;    utiv;  tiuv;    tiuv; 

vuu;    vuti;  vuu;    vuu" 

uvu;    uvu;  uvu;    uvti 

u+l,  u+t,  v;  u-Hl,  ^-u,  V 

v+i,  u+5,  u;  |-v,  u+§,  u 

u+l,  v-l-iu;  ^-u,  i-v,  u 


(48e) 


u+l,  u,  v+l 
v+l,  u,  u-hl 
U+5,  V,  u-|-§ 
u,  u-hi,  v-i-| 
V,  u+l,  u+^ 
u,  v-h^,  u-f-^ 
uOO 
OuO; 
OOu; 
uOO; 
OuO; 
OOu: 

nJ_i     1      -1- 


|-v,  u,  l-u 
^-u,  v,u-}-i 
u,  l-u,  l-v 
V,  u-hl,  ^-u 
u,^-v,  u+l 


§-u,  u+iv;   ^-u,  ^-u,v 

4  —  \T    i  — 11    n*     v-4-i     i  — 11    ii 


_     V,  |-u,  u;  v-f-i,  |-u,  u 

u+l,  f-v,  u;  ^-u,  v-M,  ii 

|-u,  u,  |-v;  l-u,  u,  v-l-l 

|-v,  ti,  u+i;  v+l,  u,  |-u 

'     u;  l-u,  V,  l-u 

V,  ^-u,  l-u 


u-Fi  V,  ^ 


11,-,. 
2  2  *i> 

uO^; 
Oui; 


.    -,  cr,  ^-u,  \ 

\,  0,  l-u;  0,  i  ^-u 

u^O;  u+i  0,  0 

0,  u-f-l,  0;  HO; 


SPECIAL   CASES   OF   THE   CUBIC   SPACE-GROUPS. 


113 


FORTY-EIGHT  Equivalent  Positions.— Confmucd. 

1,  h   u+l;  0,  0,  u+i;  0|u;   ^Ou; 


2  >  2    ^J  2  > 


(48f) 


2)   5>  2"~u;  0,  0,  2 


(48g) 


(48h) 


utiO 
Ouu 
uOu 
uuO 
Ouu 
uOu 
uuO 
Ouu 
uOu 
uuO 
Ouu 
uOti 
uu^ 
|uu 
u|u 
uu| 
|uu 
u^u 
uu^ 
|uu 
u|u 
uui^ 
|uu 
u^u 


u-l-i  i-u,  0; 
i  u+i  u 
l-u,  I,  u 
u+i  u+i 
I,  u-l-i  u 
u+i  I,  u 
u,  l-u 


1 

2 

i  |-u,  u 
^-u,  i,  u 
|-u,  u-l-l 
i,   §-u,  u 

u+^,  ^-u 
0,  u+l,  u 
§— u,  0,  u 
u-fl,  u-f-| 
0,  u+l,  u 
u+i  0,  u 
|-u,  ^-v 
0,  ^-u,  u 
|-u,  0,  u 
|-u,  u-M 
0,  ^-u,  u 
u+^,  0,  u 


0,  i-u,  0;  iuO; 
tiiO;   l-u,  0,  0; 
0|u;   |0u. 

2>  u,  ^— u; 


0,  u+§,  l-u; 


0; 


l-u,  0,  u+i;  u,  i  u+^; 

u+l,  u,  ^;  u,  u+i  ^; 

I,  u,  u+l;  0,  u-l-^,  u+l; 

u+l,  0,  u-\-i;  u,  i  u+§; 


0; 


0; 


1 . 

2  > 


1. 

2  > 


1. 
2  > 


1  . 

2  > 


2— U,  U,  ^, 


u,  ^-u,  ^; 


I,  u,  i-u]         0,  ^-u,   |-u; 
i-u,  0,  |-u;  u,  i  ^-u; 
^-u,  u,  I;    u,  u+l,  I; 

0,  ^-u,  u+§; 


u,  f-u,  0; 

1  ii_i_i  1  11- 
1)   u-f-^,  5  — u, 


u,  t-u,  t; 
i  u,  i-u; 
4-u,  i  u; 
u+i  f-u,  i; 


u,  u+f,  I; 
I,  u,  u+f; 
u+f,  h  u; 
u+l,  u+i  I; 


i  u,  u+§; 
u+i  0,  ^-u; 
u+i  Q,  0; 
0,  u,  |-u; 

i-u,  i  u+l;  u,  0,  u+i; 

u+l,  u,  0;  u,  u+l,  0; 

0,  u,  u+l;  I,  u+l,  u+l; 

u+i  i  u+l;  u,  0,  u+^; 

i-u,  u,  0;  u,  ^-u,  0; 

0,  u,  ^-u;  i  l-u,  ^-u; 

^-u,  h  i-u;  u,  0,  |-u; 

^-u,  u,  0;  u,  u+^,  0; 

0,  u,  u+l;  i  ^-u,  u+l; 

u+i  i  l-u;  u,  0,  l-u. 

u,  u+f,  1; 


i  u,  u+f; 
u+f,  h   u; 
^-u,  u+i,  i; 
8>  ^~u,  u+4; 
u+i  I,  u; 
u+i  i-u,  t;  u+l,  u+f,  I;  ^-u,  f-u,  f ;  |-u,  u+f,  f ; 


i  u+l,  f-u;  f,  u+iu+f; 
u+f,  f,  u; 


f-u, 


f,  u; 


u,  i-u,  I; 
I,  u,  f-u; 
f-u,  I,  u; 

2    U,  4   U,  8, 

ii-u,  f-u; 
l-u,  I,  ti; 


i  u,  f-u; 


f,  u,  u+f; 


f,  u,  f-u;   f,  u,  u+f; 

f-u,  iu+^;  u+f,  I,  u+l;  f-u,  |,|-u;  u+f,i|-u; 

u,  f  -u,  I; 


u,  u+f,  I;   u,  f-u,  t;   u,  u+f,  f; 


iu+if-u;  I,  u+iu+f;  |,  f 


•u;  i  l-u,  u+f; 


(48i) 


f-u,  I,  u+l;  u+f,  I,  u+l;  f-u,  f,  |-u;  u+f,  f,  |-u. 

fOO;  f|0;  fO|;  ii 

f  0  0; 

Of  0; 

OfO:  IfO;  HI; 


4  2^,    4  "  2  , 

i  3  n-   i  1  !• 
2  4  ">    2  ¥  2  I 


4 1 1; 

3  1  !• 

4  2  2, 

n  3  1 . 

'-'4  2. 


114 


SPECIAL   CASES   OF   THE    CUBIC   SPACE-GROUPS. 


FORTY-EIGHT  Equivalent  Positions.— Con^wwed. 


OOi; 

111. 
2  2  4; 

HI;  OH; 

OOf; 

113. 
2  2  4; 

Hi;  OH; 

Hi; 

3  11. 

4  2  4; 

3  n  3.   113. 

4  '-'  4;   4  2  4; 

Hi; 

3  0  1  • 
4^4; 

3  13.  1  n  3  . 

4  2  4;   4  "  4  ; 

'J  4  4; 

13  1. 
2  4  4; 

113.   n  3  3  . 
2  4  4;   'J  4  4; 

111. 
2  4  4; 

nil. 

"J  4  4; 

All-   13  3. 
'-'4  4;   2  4  4; 

i|0; 

3  3  f). 

4  4^; 

3  11.    13  1. 

4  4  2;   4  4?; 

111. 

4  4  2; 

3  3  1. 

4  4  2; 

3  1  n.   1  3  n 

4  4  U,   4  4  U. 

(48j) 

Ouv 

Ouv; 

i  u+i  v+^; 

i  u+i  Hv; 

vOu 

vOu; 

v+i  i  u+l; 

l-v,  i  u+l; 

uvO 

uvO; 

u+l,  v+i  1; 

u+i  l-v,  ^; 

uOv 

aOv; 

1    nil    ,,• 
2""U,  2;  2~V, 

^-u,  i  v+^; 

Ovu 

Ovu; 

i  l-v,  ^-u; 

i  v+i  |-u; 

vuO 

vuO; 

i-y,   Hu,  1; 

v-l-i  1  —  11  i- 
V-r2;  2   ^)    2; 

Ouv 

Ouv; 

i  l-u,  Hv; 

1    1    n   ir-Ll- 

2,  2— U,  V-|-2, 

vOu 

vOu; 

i-v,  i  l-u; 

v+l,  i  |-u; 

uvO 

uvO; 

Hu,  Hv,  1; 

|-u,  v+l,  ^; 

uOv 

uOv; 

u+i  i  v+i; 

u+i  i  i-y; 

Ovu 

Ovu; 

i  v+i  u+^; 

i,  l-v,  u+l; 

vuO 

vuO; 

v+i  u+i  1; 

^-v,  u+i,  1; 

(48k) 

UU  V 

;  u  u  V 

;  u+i  u+i  v+l; 

i-u,  u+l,  v+^; 

vuu 

;  vuu 

;  v+i  u+i  u+i; 

v+l,  Hu,  u+l; 

u  vu 

;  u  V  u 

;  u+i  v+i  u+l; 

u+i  v+l,  Hu; 

u  ti  V 

;  u  u  V 

;  Hu,  Hu,  l-v; 

l-u,  u+i  Hv; 

u  vu 

;  u  V  ti 

;  Hu,  l-v,  l-u; 

u+i  l-v,  |-u; 

vuu 

;  vtiu 

;  i-v,  ^-u,  l-u; 

l-v,  Hu,  u+l; 

UU  V 

;  UU  V 

;  u+i  l-u,  Hv; 

Hu;  l-u,  v+l; 

vuu 

;  vuu 

;  i-v,  u+i  l-u; 

v+i  Hu,  l-u; 

U  V  u 

;  ti  vti 

;  l-u,  |-v,  u+l; 

|-u,  v+i  |-u; 

uti  V 

;  uu  V 

;  u+i  Hu,  v+l; 

u+i  u+i  |-v; 

u  vu 

;  u  V  u 

;  ^-u,  v+i  u+^; 

u+l,  |-v,  u+l; 

vuti 

;  vuu 

;  v+l,  u+l,  ^-u; 

|-v,  u+l,  u+|. 

(481) 

u,  u- 

j_i  1  • 

r2;  4; 

u,  ^-u,  i;  ti,  u+i 

3.   f,   1_,,   1. 
4 ;   ";  2   i^;  4 ; 

i  u, 

u+^; 

f;  u,  ^-u;  i  u,  u+l;  i  u,  ^-u; 

u+i 

h  u; 

^-u,  f,  u;  u+i  i 

u;  l-u,  i,  a; 

u+i 

u,  i; 

^-u,  u,  1;  u+l,  ti, 

f;  l-u,  ti,  i; 

u,  i 

u+i; 

u,  i  ^-u;  u,  f,  u+^;  u,  I,   |-u; 

iu^ 

hi  u; 

f,  i-u,  u;  f,  u+i 

u;  i  i-u,  u; 

u,  ^- 

-u,  f; 

u,  u+l,  i;  u,  l-u. 

i;  u,  u+l,  1; 

f ,  u, 

l-u; 

i,  ti,  u+l;  i,  u,  1- 

-u;  i  u,  u+l; 

^-u, 

f;  ti; 

u+i  i  ti;  ^-u,  i, 

u;  u+l,  f,  u; 

^-u, 

U;  f; 

u+i  ti,  i;  |-u,  u, 

i;  u+l,  u,  f; 

u,  f, 

l-u; 

u,  i,   u+l;  u,  i,  ^- 

-u;  u,  f,  u+l; 

f,  ^ 

-u,  u; 

i  u+i  ti;  i  ^-u, 

u;  f,  u+l,  u; 

SPECIAL   CASES   OF   THE    CUBIC   SPACE-GlROtJPS.  Il5 

FORTY-EIGHT  Equivalent  Positions.— Con^mwed. 


(48m)  uO|; 

l,_Li      1       3, 

l-u,  0,  f; 

,-,11. 

"2   4; 

iuO; 

i  u+l,  i; 

i  l-u,  0; 

|u|; 

Oiu; 

i  i  u+l; 

0,  i  l-u; 

§ia; 

i  i-u,  0; 

3       3_,,       1. 
4>     4         ^>     2  } 

i  u+i  1; 

i  u+i  0 

i-u,  0,  h 

3_„       1        3. 

4         ^)     2)     4} 

u+i  h  i; 

u+i  0,  f 

0,  h  i-u; 

1       3       3_,,. 

2)     4>     4        l^; 

1       1       11-4-3. 
2j     4>     LI    14? 

0,  i  u+i 

tiOf; 

i  — 11      A      i- 
2        ">     2>     4; 

„  1   3. 
U  2    4; 

u+i  0,  i 

|Q0; 

i  l-u,  1; 

fu|; 

i  u+i  0 

Ofti; 

i  i  ^-u; 

Hu; 

0,  i  U+I 

h  u+i,  1; 

1,  u+f,  0; 

i  f-u,  0; 

3  1         „       1 

4  J     4— U,    ^ 

u+i,  i  h 

u+i  0,  f; 

f-u,  0,  I; 

i  — 11      i      ^ 
4         ^}     2>     i 

i  i  u+i; 

0,  f,  u+l; 

0,  i  f-u; 

1       3       1_,, 

2>     4>     4         U 

(48n)  u,  u+i  i; 

%  i-u,  i; 

u,  f-u,  f; 

u,  U+i  f 

i,  u,  u+J; 

i  u,  i-u; 

i  u,  f-u; 

i  u,  U+f 

u+i  i  u; 

i-u,  i  u; 

f-u,  i  u; 

u+i  i  a 

u+i  i-u,  I;  l-u,  u+i  I;  u+i  u+i  f ;  |-u,  f-u,  f ; 

8,  u+2,  4— u;  8,  2— u,  u+j;  ■§,  u+^,  u+4;   g,  2— u,  4— u; 

4  — u,  8)  u+2;  u+4,  8,  2— u;  u+4,  8>  u+2;  4— u,  g,  2— u; 

u,  f-u,  I;  u,  u+i  f;  u+i  f-u,  i;    u,  u+f,  |; 

i  u,  f-u;  i  u,  u+i  i  u+i  f-u;    i  u,  u+f; 

f-u,  I,  u;  u+i  f,  u;  f-u,  i  u+|;   u+i  |,  u; 

l-u,  i-u,  I;  |-u,  u+f,  i;  u,  i-u,  f ;        u+i  u+i  f; 

i|-u,  j-u;  i,  l-u,  u+f;  i  u,  f-u;        i  u+i  u+f ; 

f-u,  i  i-u;  u+ii|-u;  f-u,  i  u;        u+f ,  i  u+i 

SIXTY-FOUR  Equivalent  Positions. 

(64a)  uuu;  uuti;  uuu;  tiuu; 
tiuu;  tiuu;  uuu;  uuu; 
|-u,  |-u,  |-u;    |-u,  u+i  u+i    u+i  i-u,  u+i 

u+i  u+i  i-u; 
u+i  u+i  u+i    u+i  l-u,  i-u;    l-u,  u+i  |-u; 

l-u,  |-u,  u+i 
u+i  u+i  u;   u+i|-u,  u;   ^-u,  u+iu;   |-u,  |-u,  u; 
|-u,  |-u,  u;    |-u,  u+iu;   u+i|-u,  u;   u+i  u+i  u; 
u,  u,  |-u;        u,  u,  u+i        u,  u,  u+i        u,  u,  |— u; 
u,  u,  u+i        u,  u,  |-u;        u,  u,  i-u;        u,  u,  u+i 
u+i  u,  u+i    u+iti,  |-u;    |-u,  u,  |-u;    |-u,  u,  u+i 
|-u,  u,  i--u;    |-u,  u,  u+i    u+i  u,  u+i    u+i  u,  |-u; 
u,  |-u,  u;        u,  u+i  u;       u,  |-u,  u;        u,  u+i  u; 
u,  u+i  u;        u,  |-u,  u;       u,  u+i  u;        u,  |-u,  u; 
u,  u+i  u+i    u,  |-u,  §-u;    u,  u+i|-u;    u,  |-u,  u+i 
u,  l-u,  |-u;   u,  u+i  u+i   u,  l-u,  u+i   u,  u+i|-u; 
|-u,  u,  u;        |-u,  u,  u;        u+i  u,  u;       u+i  u,  u; 
u+i  u,  u;        u+i  u,  u;        |-u,  u,  u;       |-u,  u,  u. 


116 


SPECIAL   CASES  OF  THE   CUBIC  SPACE-GROUPS. 


SIXTY-FOUR  Equivalent  Positions. — Continued. 

(64b)  uuu;    uuu;    uuti;    utiu; 

l-u,  i-u,  i-u;    u+i  J-u,  u+i;    |-u,  u+J,  u+i; 

u+i,  u+l,  J-u 
f-u,  f-u,  f-u;    l-u,  u+f,  u+f;    u+|,  f-u,  u+f; 

u+f,  u+f,  f-u 
u+i,  u+l,  u+l;    |-u,  u+l,  |-u;    u+^,  |-u,  ^-u; 

^-u,  ^-u,  u+i 
u+^,  u+l,  u;  u+l,  §— u,  u;  |-u,  u+§,  u;  |-u,  |-u,  u 
f-u,  f-u,  i-u;    u+f,  f-u,  u+i;    f-u,  u+f,  u+f; 

u+f,  u+f,  f-u 
f-u,  f-u,  f-u;    f-u,  u+f,  u+f;    u+f,  f-u,  u+f; 

u+f,  u+f,  f-u 
u,  u,  u+l;        u,  u,  ^-u;       u,  u,  |-u;        u,  u,  u+|; 
u+l,  u,  u+l;   u+l,  u,  ^-u;   |-u,  u,  ^-u;   |-u,  u,  u+| 
f-u,  f-u,  f-u;    u+f,  f-u,  u+f;       f-u,  u+f,  u+f 

u+f,  u+f,  f-u 
f-u,  f-u,  f-u;    f-u,  u+f,  u+f;    u+f,  f-u,  u+f; 

u+f,  u+f,  f-u 
u,  u+l,  u;       u,  u+l,  u;        u,  §-u,  u;       u,  |-u,  u; 
u,  u+^,  u+l;   u,  ^-u,  |-u;   u,  u+|,  |-u;   u,  ^-u,  u+| 
f-u,  f-u,  f-u;    u+f,  f-u,  u+f;       f-u,  u+f,  u+f 

u+f,  u+f,  f-u 
f-u,  f-u,  f-u;    f-u,  u+f,  u+f;    u+f,  f-u,  u+f; 

u+f,  u+f,  f-u 
u+l,  u,  u;    |-u,  u,  u;    u+|,  u,  u;    |— u,  u,  u. 

NINETY-SIX  Equivalent  Positions. 


(96a)  Ouv 

Ouv 

Ouv 

Ouv; 

vOu 

vOu 

vOu 

vOu; 

uvO 

uvO 

uvO 

uvO; 

uOv 

uOv 

uOv 

uOv; 

Ovu 

Ovu 

Ovu 

Ovu; 

vuO 

vuO 

vuO 

vuO; 

h  u+i  v; 
v+i  I,  u; 


1    1  —11 

1)     2        U 


v; 
l-v,  i  u; 


h  u+l,  v; 
2-v,  I,  u; 


h,  l-u,  v; 
v+l,  I,  u; 


u+l,  v+l,  0;  l-u,  |-v,  0;  u+|,  |-v,  0;  |-u,  v+|,  0; 

|-u,  I,  v;  u+l,  I,  v;  |-u,  |,  v;  u+|,  0,  v; 

I,  |-v,  H;  I,  v+l,  u;  |,  v+|,  u;  |,  |-v,  u; 

|-v,  l-u,  0;  v+l,  u+l,  0;  v+|,  |-u,  0;  |-v,  u+|,  0; 

I,  u,  v+l;  I,  u,  |-v;  |,  u,  |-v;  |,  u,  v+|; 

v+l,  0,  u+l;  l-v,  0,  |-u;  |-v,  0,  u+|;  v+|,  0,  |-u; 

u+l,  V,  I;  l-u,  V,  I;  u+|,  v,  |;  |-u,  v,  |; 

l-u,  0,  |-v;  u+l,  0,  v+l;  |-u,  0,  v+|;  u+|,  0,  |-v; 

I,  V,  u+l;  I,  V,  l-u;  |,  v,  u+|; 


I,  V,  l-u; 


SPECIAL   CASES   OF  THE   CUBIC   SPACE-GROUPS. 


117 


NINETY-SIX  Equivalent  'Posttio^s.— Continued. 


^-v,  u,  I;        v+l,  u,  i; 
0,  u+iv-1-^;   0,  l-u,  |-v; 
V,  h  u+l;        V,  i  ^-u; 


u,  v-l-l,  ^; 
ti,  i  ^-v; 
0,  ^-v,  ^-u 
V,  l-u,  I; 
(96b)  uuv;  utiv 
vuu;  vuu 
uvu;  tivu 
utiv;  uuv 
uvu;  uvu 
vuu;  vuu 
u+i  u+^,  V 
v+iu-l-iu 
u-h^,  v-l-^,  u 
^-u,  ^-u,  V 
l-u,  ^-v,u 
l-v,  i-u,u 
u+l,  u,  v+l 
v+l,  u,  u+l 
u-}-|,  V,  u+l 
i-u,  u,  |-v 


1-u,  V,  l-u 
^-v,  u,  |-u;  v+^,  u,  |-u 
u,  u+iv+^;  u,  i-u,  i-v 
v,u-H|,  u+l;  V,  u+l,  i-u 
u,  v+^,  u+^;  u,  |-v,  u+l 
u,  ^-u,  i-v;  u,  ^-u,  v+l 
u,  ^-v,  |— u;  Q,  v+^,  u+^ 
^^,  ^-u,  i-u;  V,  u+ii-u 
(96c)  uuO;  u+i  u,  i; 
Ouu;  ^,  u,  u-f-|; 
uOu;    u+^,  0,  u+l; 


u,  i-v,  i; 
u,  i  v+l; 
0,  v+i  u+l; 
V,  u+i  I; 
utiv 
vtiii 
ti  V  ti 
uuv 
uvu 
vuu 
u+t,  t-u,  V 
i-v,  u+iti 
^-u,  |-v,u 
u+i  |-u,  V 
|-u,  v+iu 
v+iu+iti 
u+i  u,  i-v 
|-v,  u,  i-u 
^-u,  V,  U+I 
u+i  u,  v+l 

|-U,  V,  U+I 


v+i  u,  I;  l-v,  u,  I; 

0,  u+i|-v;  0,  l-u,  v+l; 

V,  i  u+i  V,  i  l-u; 

u,  l-v,  i  ti,  v+i  i- 

u,  i  v+i  u,  i  |-v; 

0,  v+i  |-u;  0,  l-v,  u+i 

V,  l-u,  i  V,  u+i  i 


uuv 
V  tiu 
u  V  ti 
tiu  V 
u  V  ti 
vtiu 


atiO;    |-u,  u,  i 


1 . 

l-u,  0, 
1)  u,  ^- 

1 

2 


tiOti; 
0  ti  ti; 
l-u,  l-u 
i  l-u,  l-u 
l-u,  i  l-u 
u+i  u+i  I 
u+i  i  U+I 
i  u+i  U+I 


l-u; 

u; 

ti,  |-u,  0; 

0,  l-u,  ti; 

u|u; 

u,  u+i  0; 

u|u; 

0,  u+i  u; 


|-u,  u+iv;  |-u 
l-v,  |-u,u 


u+i  |-v,ti 
l-u,  u+iv 
u+i  v+i  u 
v+i|-u,  u 
l-u,  u,  l-v 
|-v,  U,  U+I 
u+i  V,  l-u 
l-u,  U,  V+I 
u+i  V,  l-u 
V+I,  ti,  U+I 
u,  u+i  l-v 
V.  |-u,  U  +  I 
u,  l-v,  l-u 


u,  u+i  V+I 

u,  v+i  l-u 

V,  l-u,  U+I 

uuO 

Ouu 

uOu 

uuO 

Ouu 

uOu 

|— u,  U+I,  I 

i  l-u,  U+I 

u+i  I,  |— u 

u+i  l-u,  I 

i  u+i  l-u 

|— u,  i  U+I 


1 

)    2 


U,  V 

V+I,  l-u,  u 
l-u,  v+i  u 
u+i  u+i  V 
u+i  |-v,u 
l-v,  u+i  u 
l-u,  U,  V+I 
v+i  ti,  l-u 
l-u,  V,  l-u 
u+i  u,  |-v 

U+I,  V,  U+I 
l-v,  U,  U+I 


'-'  i-u,v  +  | 

1 


U,  2 

1 


V,  t-u,  f-u 
u,  V+I,  l-u 
u,  U+I,  l-v 
u,  l-v,  U+I 
V,  u  +  i  U  +  I 

u+i  ti,  I; 
I,  u,  l-u; 
l-u,  0,  U+I; 
l-u,  u,  I; 
I,  ti,  U+I; 
u+i  0,  l-u; 

ti,  u+i  0: 

0,  |-u,  u; 

u|u; 

u,  |-u,  0; 

0,  u+i  ti; 


u|u; 


118 


SPECIAL   CASES   OF   THE   CUBIC   SPACE-GROUPS. 


NINETY-SIX  Equivalent  Positions. — Continued. 

M+h  u-l-l,  0;    u,  u+i  I;    u+|,  |-u,  0;    u,  i-u,  |; 

i  u+^,  u;    0,  u+i  i-u; 
l-u,  i,  u;    u,  J,  u+l; 
l-u,  u+l,  0;    u,  u+l,  ^; 
i  i-u,  u;    0,  ^-u,  u+l; 

uu§;  |-u,  u,  0; 

0,  u,  u+§;     |uu; 

u,  0,  |-u;    u-}-|,  0,  u; 


i  u+i  u;    0,  u+l,  u+l; 
u+i  i  u;    u,  i  u+l; 
|-u,  l-u,  0;    u,  |-u,  i; 

^-u,  u,  0; 


2  u>  2>  u; 

2»  2~u,  u; 

0,  u,  ^-u; 

u,  0,  |-u; 


l-u,  0,  u; 

u+i  u,  0; 
u,  0,  u+l;  u+l,  0,  u; 
0,  u,  u+l;    ^uu; 


(96d)  uvi; 
iuv; 


2         "?     2         V,     4 

1 


vuf; 
ufv; 
f  vQ; 
uvf; 
f  uv; 


u 

iv; 

1 

4 

vu; 

uvf; 

f 

uv; 

vfu; 
vui; 
ujv; 
?vu; 
uvi; 
iuv; 
viu; 


i-v,  i  l-u 

v+i  u+i  I 

u+i  i  v+i 

I,  v+i  u+i 

l-u,  v+i  f 

f,  l-u,  v+l 
vf  u;  v+i  f,  l-u 
vui;    |-v,  u+i  i 

u+i  i  ^-v 

i  |-v,  u+i 

i  u+i  l-v 

i-v,  f,  u+l 

v+i  i-u,  i 

^-u,  i  v+l 

i,  v+i  ^-u 

u+i  v+i  i 

i  u+l.  v+l 

v+i  i,  U+I 
vuf;  l-v,  i-u,  I 
ufv;  l-u,  f,  l-v 
f  vu;  f,  |-v,  l-u 
(96e)  uuv;  utiv;  uuv;  u  ti  v; 
vuu;  vuu;  vuu;  vuu; 
uvu;  uvu;  uvti;  uvti; 
l-u,  i-u,  f-v;    u+f,  f-u,  v+i;    f-u,  u+f,  v+|; 

u+i  u+f,  f-v; 
f-u,  i-v,  l-u;    f-u,  v+i  u+i;    u+f,  v+i  f-u; 

u+i  i-v,  u+i 


utii' 

u+i  u,  0; 

0,  u,  i- 

■u, 

|uu; 

u,  0,  u+i 

|-u,  0,  u. 

i  u+i 

V 

;  i  u,  l-v; 

v+i  f, 

u 

;  V,  i  U+I; 

i-u,  i 

V 

;  u,  f,  V+I; 

i  l-v, 

u 

h   V,  l-u; 

i  u+i 

V 

;  i  u,  V+I; 

l-v,  i 

u 

;  V,  f,  U+I; 

l-u,  i 

V 

u,  i  ^-v; 

i  v+i 

u 

h  V,  l-u; 

1   1   n 
4>  2   U, 

V 

u+i  V,  f ; 

v+i  i 

u 

l-v,  u,  i 

u+i  f, 

V 

u+i  V,  i; 

f,  l-v, 

u 

v+i  u,  f ; 

3   1_„ 

4j  2    ^} 

V 

l-u,  V,  f; 

h-y,  I 

u 

l-v.  u,  f; 

u+i  i 

V 

l-u,  V,  f; 

i  v+i 

u 

v+i  u,  f; 

f,  u,  i- 

•V 

u,  l-v,  f; 

■Tr    3    1 

V,  i,     2- 

-u 

V,  u+i  I; 

u,  i,  V+I 

u,  v+i  f; 

i  V,  U+I 

V,  u+i  f ; 

i  u,  v+i 

u,  l-v,  f; 

V,  i  1- 

■u, 

V,  l-u,  f; 

n    3    1 
U,  "4,  1- 

■V, 

u,  v+i  f ; 

f,  V,  U+I; 

u,  l-u,  f. 

SPECIAL   CASES   OF   THE   CUBIC   SPACE-GROUPS.  119 

NINETY-SIX  Equivalent  Positions. — Continued. 

i-v,  l-u,  i-u;    v+l,  u+i  i-u;    v+i,  i-u,  u+i; 

l-Y,  u+i  u+i; 
u+l,  u+l,  v;  u+i|-u,  v;  ^-u,  u+|,  v;  |-u,  |-u,  v; 
v+l,  u+iu;  l-v,  u-f-|,  u;  ^-v,  ^-u,  u;  v+|,  ^-u,  ti; 
u+iv+iu;  l-u,  l-v,  u;  u+i|-v,  u;  ^-u,  v+iu; 
f-u,  f-u,  i-v;    u+f,  f-u,  v+i;     f-u,  u+f.  v+|; 

u+f,  u+f,  T-v; 
f-u,  f-v,  i-u;    f-u,  v+f,  u+i;    u+f,  v+f,  i-u; 

u+f,  f-v,  u+i; 
f-v,  f-u,  i-u;    v+f,  u+f,  l-u;    v+f,  f-u,  u+i; 

f-v,  u+f,  u+i; 
u+l,  u,  v+l;  u+iu,  ^-v;  |-u,  u,  |-v;  ^-u,  ti,  v+|; 
v+iu,  u+l;  l-v,  u,  ^-u;  |-v,  u,  u+^;  v+i  ti,  i-u; 
u+^,  V,  u+l;  ^-u,  V,  u+^;  u+|,  v,  ^-u;  ^-u,  v,  ^-u; 
f-u,  i-u,  f-v;    u+f,  i-u,  v+f;    f-u,  u+i,  v+f; 

u+f,  u+i  f-v; 
f-u,  l-v,  f-u;    f-u,  v+i  u+f;    u+f,  v+i  f-u; 

u+f,  i-v,  u+f; 
f-v,  i-u,  f-u;    v+f,  u+f,  f-u;    v+f,  i-u,  u+f; 

f-v,  u+f,  u+f; 
u,  u+iv+§;  u,  ^-u,  ^-v;  u,  u+i|-v;  u,  ^-u,  v+^; 
V,  u+l,  u+l;  V,  u+l,  |-u;  v,  ^-u,  u+^;  v.  ^-u,  |-u; 
u,  v+iu+l;  u,  l-v,  u+^;  u,  |-v,  ^-u;  u,  v+i  |-u; 
f-u,  f-u,  f-v;    u+f,  f-u,  v+f;     f-u,  u+f,  v+f; 

u+f,  u+f,  f-v; 
f-u,  f-v,  f-u;    f-u,  v+f,  u+f;    u+f,  v+f,  f-u; 

u+f,  f-v,  u+f; 
f-v,  f-u,  f-u;    v+f,  u+f,  f-u;    v+f,  f-u,  u+f; 

f-v,  u+f,  u+f. 
(96f)  u,  f-u,  i;    u,  u+f,  I;    u,  f-u,  |;    u,  u+f,  i; 
h   u,  f-u;   I,  u,  u+f;   |,  u,  f-u;   i  u,  u+f; 
f-u,  i,  u;    u+f,  I,  u;    f-u,  i  u;    u+f,  i  u; 
u+i  f-u,  i;  u+i  u+f,  I;  |-u,  f-u,  |;  |-u,  u+f,  i; 
i  u+i  f-u;  I,  u+i  u+f;  f,  ^-u,  f-u;  f,  i-u,u+f; 
f-u,  f,  u;    u+f,  I,  u;    f-u,  f,  u;    u+f,  |,  u; 
u+i  f-u,  f;  u+i  u+f,  f;  ^-u,  f-u,  |;  ^-u,  u+f,  f; 
f,  u,  f-u;    f,  u,  u+f;    f,  u,  f-u;    f,  u,   u+f; 
f-u,  i  u+i  u+f,  I,  u+i  f-u,  i  ^-u;  u+f,  i|-u; 
u,  f-u,  f;   u,  u+f,  t;   u,  f-u,  f;   u.  u+f,  |; 
i  u+i  f-u;  I,  u+i  u+f;  |,  |-u,  f-u;  i|-u,  u+f; 
f-u,  f,  u+i  u+f,  I,  u+i-  f-u,  I,  l-u;  u+f,  f,  ^-u; 
f-u,  u,  i;   u+f,  u,  I;    f-u,  u,  |;    u+f,  u,  |; 
u,  i  f-u;   u,  I,  u+f;   u,  i  f-u;   u.  f.  u+f; 


120 


SPECIAL   CASES  OF  THE   CUBIC  SPACE-GROUPS. 


NINETY-SIX  Equivalent  Positions. — Continued. 


h  i-u,  u; 
f-u,  u+i  I; 

U+2>    8>   4        Uj 


h  u+f,  u; 
u+i  I,  u+f  ; 


h  i-u,  u; 
l-u,  ^-u,  I; 
l-u,  f,  i-u; 


I,  u+f,  u; 
l-u,  I,  u+f; 


1,  l-u,  u; 

i  u+f,  u; 

i  f-u,  u; 

i  u+f,  u; 

l-u,  u+il; 

u+f,  u+l,  f ; 

f-u,  ^-u. 

f; 

u+f,  i-u,|; 

u,  f,  f-u; 

u,  1,  u+f; 

u,  i  f-u; 

%  I  u+f; 

1,  f-u,  u+l; 

1,  u+f,  u+i; 

h  f-u,  |  — 

u; 

iu+f,|-u; 

f-u,  u,  f ; 

u+f,  u,  1; 

f-u,  u,  1; 

u+f,  u,  1; 

u+i  1,  f-u; 

u+i  1,  u+f; 

i-u,  1,  f- 

u; 

l-u,  iu+f; 

f,  f-u,  u+l; 

f,  u+f,  u+l; 

3      1         „      1 
¥,   4— U,   2  — 

u; 

i  u+f,|-u. 

(96g)  uOO; 

u+l,  h  0; 

u+i  0,  1; 

u  2  2 , 

tiOO; 

^-u,  1,  0; 

^— u,  0,  ^; 

uH; 

OuO; 

h  u+i.  0; 

^u|; 

0,  u+l,  A; 

OuO; 

i  ^-u,  0; 

Hi; 

0,  i-u,  1; 

OOu; 

Hu; 

i  0,  u+l; 

0,  1,  u+l; 

OOu; 

Hu; 

2,  0,  i-u; 

0,  i,  h-u; 

h  f-u,  f ; 

f,  f-u,  f; 

3       1_,,       3. 
4>     4        ",     4, 

h  f-u,  f; 

f,  u+f,  f ; 

3      1,4.3      1. 
4>     '^14,     4, 

3      „4_1       3. 
4>     "T^4,     4, 

h  u+f,  f ; 

4        U>    4,    4> 

3_,,       3       1. 
4        11,     4,     4> 

3  „       1       3  . 

4  — U,     4j     4, 

i_ii    3    3. 

4         ^)     4>     4> 

U+f,  f,  f ; 

il_J_3       3       1. 
^lif     4,     4, 

U+f,  f,  f ; 

U+f,  f,  f ; 

i,  h  i-u; 

f,  i  f-u; 

3       1       3_„. 

4>     4>     4        >^, 

1      3       3_,,. 

4>     4>     4        ", 

i  i  u+f; 

f,  f,  u+f; 

f,  h  U+f; 

h  f,  U+f; 

f-u,  f,  f; 

f-u,  f,  f; 

f-u,  f,  f; 

f-u,  f,  f; 

u+f,  f,  f ; 

u+f,  f,  f ; 

u+f,  f,  f; 

u+f,  f,  f; 

f,  f-u,  f; 

h  f-u,  f; 

f,  f-u,  f; 

f,  f-u,  f; 

f,  u+f,  f ; 

h  u+f,  f ; 

f,  u+f,  f; 

i  u+f,  f ; 

3       3       3_,,. 
4>     4,     4        "> 

f,  f,  f-u; 

h  f,  f-u; 

3    1    i_„. 

4>     4>     4        ", 

f,  f,  U+f; 

1      1      ii_L3. 

4>     4,     U-r4, 

f,  i  u+f; 

f,  f,  U+f; 

h  u+i,  1; 

Oui; 

0,  u+^,  0; 

iuO; 

2)    3       U,    2J 

OQA; 

0,  i-u,  0; 

itiO; 

u+l,  h  ^; 

uOJ; 

uiO; 

u+l,  0,  0; 

l-u,  i  i; 

uO|; 

QH; 

l-u,  0,  0; 

h  h,  u+l; 

0,  0,  u+l; 

O^u; 

|0u; 

i  i  l-u; 

0,  0,  i-u; 

0|u; 

|0u; 

(96h)  u,  f-u,  J; 

u,  u+f,  1; 

u,  f-u,  1; 

u,  u+f,  i; 

i  u,  f-u; 

1,  u,  u+f; 

1,  u,  f-u; 

i  u,  u+f; 

f-u,  i  u; 

u+f,  h  u; 

f-u,  1,  u; 

u+f,  i  u; 

u+if-u,  i; 

u+i  u+f,  1; 

i-u,  f-u, 

7 . 

8j 

l-u,  u+f,  1; 

1,  u+l,  f-u; 

f,u+|,  u+f; 

ii-u,f- 

u; 

f,|-u,u+f: 

f-u,  f,  u; 

u+f,  1,  u; 

f-u,  f,  u, 

u+f,  1,  u; 

u+i  f-u,|; 

u+i  u+f,  1; 

i-u,  f-u. 

1; 

|-u,u+f,  t; 

i  u,  f-u; 

f,  u,  u+f; 

i  u,  f-u, 

i  u,  u+f; 

f-u,  |,u+^; 

u+f,  |,u+|; 

f-u,  1,1- 

u; 

u+f,  il-u; 

THE   CUBIC   SPACE-GROUPS  T -T.^  121 

NINETY-SIX  Equivalent  Positions. — Continued. 

u,  l-u,  f;  u,  u+i,  I;  u,  f-u,  |;  u,  u+i,  f; 

iu+if-u;  I,  u+iu+i;  |,  |-u,  f-u;  ii-u,u+i; 

i-u,  |,u-f-|;  u+f,  f,u+i;  i-u,  f,  §-u;  u+f,  f,^-u; 

f-u,  u+l,  I;  u+iu+l,  I;  f-u,  |-u,  f;  u+i,|-u,  |; 

u+ii  f-u;  u+it,u+i;  |-u,  |,f-u;  |-u,f,  u+i; 

I,  f-u,  u+l;  t,u+iu-f-|;  f,  f-u,  |-u;  f,u+i,^-u; 

f-u,  u,  I;  u+f,  u,  f;  f-u,  u,  f;  u+f,  u,  f; 

u,  i  f-u;  u,  I,  u+f;  u,  |,  f-u;  u,  i  u+f; 

if-u,  u+l;  I,  u+f,  u+l;  |,  f-u,  |-u;  i,u+f,|-u; 

f-u,  u,  I;  u+f,  u,  I;  f-u,  u,  |;  u+f,  u,  1; 

u+iif-u;  u+i|,u+f;  ^-u,  |,  f-u;  |-u,  iu+f; 

t,  f-u,  u;  I,  u+f,  u;  f,  f-u,  u;  f,  u+f,  u; 

f-u,  u+l,  I;  u+f,  u+il;  f-u,  |-u,  |;  u+f,  |-u,  |; 

u,  i  f-u;  u,  f,  u+f;  u,  f,  f-u;  u,  |,  u+f; 

i  f-u,  u;  I,  u+f,  u;  |,  f-u,  u;  i,  u+f,  u; 

A.  TETARTOHEDRY. 
Space-Group  T*. 

One  equivalent  position : 

(a)  la.  (b)  lb. 

Three  equivalent  positions: 
(c)  3a.  (d)  3b. 

Four  equivalent  positions: 

(e)  4a. 

Six  equivalent  positions: 

(f)  6a.  (h)  6c. 

(g)  6b.  (i)    6d. 

Twelve  equivalent  positions: 
(j)  xyz;    xyz;    xyz;    xyz; 
zxy;    zxy;    zxy;    zxy; 
yzx;    yzx;    yzx;    yzx. 

Space-Group  'P. 

Four  equivalent  positions: 

(a)  4b.  (c)  4d. 

(b)  4c.  (d)  4e. 

Sixteen  equivalent  positions: 

(e)  16a. 

Twenty-four  equivalent  positions: 

(f)  24a.  (g)  24b. 


122 


THE   CUBIC   SPACE-GROUPS   T^-T*. 


xyz;        xyz; 

zxy;        zxy; 

yzx;        yzx; 

x+i  h-y,  z; 

l-x,  y+i  z 

l-x,  |-y,  z; 

l-z,  x+iy; 

h-z,  l-x,  y 

z+i  l-x,  y; 

h-y,  ^-z,  x; 

y+i  5-z,  X 

i-y,  z+i  x; 

x+l,  y,  i-z; 

l-x,  y,  l-z 

^-x,  y,  z+l; 

l-z,  X,  l-y; 

^-z,  X,  y+i 

z+i  x,  l-y; 

i-y,  z,  x+i; 

y+i  z,  l-x 

h-y,  z,  i-x; 

X,  h-y,  l-z; 

X,  y+i  i-z 

X,  |-y,  z+l; 

z,  x+i  l-y; 

z,  i-x,  y+i 

z,  2"~x,  2~yj 

y,  ^-z,  x+i; 

y,  i-z,  ^-x 

y,  z+i  §-x. 

Space-Group  T^  (continued). 

Forty-eight  equivalent  positions: 

(h)  xyz;  xyz; 
zxy;  zxy; 
yzx;        yzx; 

x+i  y+i  z; 
z+i  x+iy; 
y+i  z+i  x; 
x+i  y,  z+i 
z+l,  X,  y+i 
y+i  z,  x+^; 
X,  y+l,  z+l; 
z,  x+i,  y+i 
y,  z+i  x+li 

Space-Group  T'. 

Two  equivalent  positions: 

(a)  2a. 
Six  equivalent  positions : 

(b)  6e. 
Eight  equivalent  positions: 

(c)  8a. 
Twelve  equivalent  positions : 

(d)  12a.  (e)  12b. 

Twenty-four  equivalent  positions : 

(f)  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy; 
yzx;        yzx;        yzx; 

x+l,  y+l,  z-}-^;    x+i  l-y,  |-z;    J-x,  y-\-^,  \-z; 

^-x,  ^-y,  z-Fl; 
z+i  x-l-l,  y-t-l;    \-z,  x-hi  ^-y;    §-z,  |-x,  y+|; 

z+i,  l-x,  l-y; 
y+i  z-j-i  x+^;    i-y,  ^-z,  x+l;    y-f-^,  J-z,  |-x; 

i-y,  z+l,  ^-x. 
Space-Group  T*. 

Four  equivalent  positions: 

(a)  4f. 

Twelve  equivalent  positions: 

(b)  xyz;  \-\-\,  ^-y,  z;  x,  y-\rh,  |-z;  ^-x,  y,  z+^; 
zxy;  z,  x+^,  |-y;  |-z,  x,  y+^;  z-F^  \-x,  y; 
yzx;    l-y,  z,  x+^;    y-f-J,  ^-z,  x;    y,  z-f-^,  ^-x. 


xyz; 
zxy; 
yzx; 


THE   CUBIC   SPACE-GROUPS   T^-l^.  123 

Space-Group  T*. 

Eight  equivalent  positions: 

(a)  8b. 

Twelve  equivalent  positions : 

(b)  12c. 

Twenty-four  equivalent  positions: 

(c)  xyz;  X,  y,  |-z;  |-x,  y,  z;  x,  |-y,  z; 
zxy;  l-z,  X,  y;  z,  ^-x,  y;  z,  x,  |-y; 
yzx;  y,  §-z,  x;  y,  z,  |-x;  |-y,  z,  x; 
x+i  y+i  z+^;    x+i  |-y,  z;    x,  y+i  |-z; 

i-x,  y,  z+^; 
z+i  x+i  y+^;    z,  x+i  ^-y;    |-z,  x,  y+§; 

z+2>  5"~x,  y; 
y+i  z+i  x+^;    ^-y,  z,  x+^;    y+^,  ^-z,  x; 

y,  z+i  ^-x. 

B.  PARAMORPHIC  HEMIHEDRY. 

Space-Group  T^. 

One  equivalent  position: 

(a)  la.  (b)  lb. 

Three  equivalent  positions: 

(c)  3a.  (d)  3b. 

Six  equivalent  positions : 

(e)  6a.  (g)  6c. 

(f)  6b.  (h)  6d. 
^tgf/i<  equivalent  positions: 

(i)  8c. 
Twelve  equivalent  positions: 
(j)  12d.  (k)  12e. 

Twenty-four  equivalent  positions: 


(1)  xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx. 

Space-Group  T^. 

Two  equivalent  positions: 

(a)  2a. 

Four  equivalent  positions: 

(b)  4d. 

(c)  4e. 

124 


THE   CUBIC   SPACE-GROUPS   T^-T^. 


Space  Group  Ti  (continued). 
Six  equivalent  positions: 

(d)  6e. 

Eight  equivalent  positions: 

(e)  8d. 

Twelve  equivalent  positions: 

(f)  12a.  (g)  12b. 
Twenty-four  equivalent  positions: 

(h)  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy; 
yzx;        yzx;        yzx; 


xyz; 
zxy; 
yzx; 


2        X,     2        Yf     2        Zj 


2        2,    f       X,     2 


-x,  y+l,  z+^;     x+l,  i-y,  z+|; 

X+2>    y+2>     2 

■y;    z+l,  l-x,  y+l;     z+i  x;+i  |-y; 


•z; 


■y,  2-z, 


2        ^) 


2— Z,    X+2,    y+2i 

y+i  z+i  l-x;    ^-y,  z+i  x+§; 

yi2)   2~Z,  x+2. 


Space-Group  T^ 

Four  equivalent  positions: 

(a)  4b.  (b)  4c. 

Eight  equivalent  positions: 

(c)  8e. 

Twenty-four  equivalent  positions: 

(d)  24c.  (e)  24a. 
Thirty-two  equivalent  positions : 

(f)  32a. 

Forty-eight  equivalent  positions: 

(g)  48a.  (h)  48b. 
Ninety-six  equivalent  positions: 


(i) 


xyz; 
zxy; 
yzx; 
xyz; 
zxy; 
yzx; 


xyz; 
zxy; 
yzx; 
xyz; 
zxy; 
yzx; 


x+i  y+i  z; 
z+i  x+i  y; 
y+h  z+i  x; 
j-x,  i-y,  z; 
i-z,  i-x,  y; 
l-y,  h-z,  x; 


xyz; 
zxy; 
yzx; 
xyz; 
zxy; 
yzx; 
x+l,  h- 


xyz; 
zxy; 
yzx; 
xyz; 
zxy; 
yzx; 
•y,  z 


l-z,  x+iy 

2~y>  2~z,  X 
l-x,  y+i  z 
z+i  ^-x,  y 
y+i  z+i  X 


2-x,  y+l,  z 
i-z,  ^-x,  y 
y+i?-z,  X 
x+l,  |-y,  z 
z+i  x+i  y 

h-y,  z+l,  X 


J-x,  l-y,  z; 
z+i^-x,  y; 
|-y,  z+i  x; 
x+J,  y+l,  z; 
l-z,  x+i  y; 
y+l,  l-z,  x; 


THE   CUBIC   SPACE-GROUPS   I?-tJ. 


125 


Space-Group  T^  (continued). 


x+i  y,  z+l; 

x+i  y,  l-z 

^-x,  y,  l-z 

l-x,  y,  z+l 

z+i  X,  y+l; 

l-z,  X,  ^-y 

l-z,  x,  y+l 

z+i  X,  l-y 

y+i  z,  x+i; 

i-y,  z,  x+l 

y+i  z,  §-x 

l-y,  z,  l-x 

^-x,  y,  i-z; 

l-x,  y,  z+l 

x+i  y,  z+i 

x+i  y,  l-z 

^-z,  X,  ^-y; 

z+l,  X,  y+l 

z+l,  X,  l-y 

l-z,  X,  y+l 

l-y,  z,  l-x; 

y+i  z,  l-x 

i-y,  z,  x+l 

y+l,  z,  x+l 

X,  y+i  z+l; 

X,  i-y,  ^-z 

X,  y+i  l-z 

X,  l-y,  z+l 

z,  x+l,  y+l; 

z,  x+i  i-y 

,     z,  i-x,  y+l 

z,  l-x,  l-y 

y,  z+i  x+l; 

y,  i-z,  x+l 

y,  i-z,  l-x 

y,  z+l,  l-x 

X,  ^-y,  ^-z; 

X,  y+i  z+l 

X,  h-y,  z+l 

X,  y+l,  l-z 

z,  l-x,  i-y; 

z,  ^-x,  y+^ 

z,  x+l,  l-y 

z,  x+l,  y+l 

y,  l-z,  ^-x; 

y,  z+i  ^-x 

y,  z+i  x+l 

y,  l-z,  x+l 

Space-Group  T^. 

Eight  equivalent  positions: 

(a)  8f.  (b)  8g. 

Sixteen  equivalent  positions: 

(c)  16b.  (d)  16c. 

Thirty-two  equivalent  positions: 

(e)  32b. 

Forty-eight  equivalent  positions: 

(f)  48c. 

Ninety-six  equivalent  positions: 

(g)  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy; 
yzx;        yzx;        yzx; 

x+l,  y+l,  z 

z+l,  x+l,  y 

y+l,  z+l,  X 
x+l,  y,  z+l 

z+l,  X,  y+l 
y+l,  z,  x+l 
X,  y+l,  z+l 
z,  X+l,  y+l 
y,  z+l,  x+l 


xyz 
zxy 
yzx 


4      X,   4      y,    4 


x+|.  l-y,  z 

l-z,  x+l,  y 
l-y,  l-z,  X 
x+l,  y,  l-z 
l-z,  X,  |-y 
l-y,  z,  x+l 
X,  |-y,  |-z 
z,  x+l,  l-y 
y,  l-z,  x+l 
z;    l-x,  y+i 


4~z,  4— X,  4— yj  z+4,  4  X, 
i-y,  i-z,  i-x;  y+i  z+|, 
l-x,  f-y,  i-z;    l-x,  y+l, 


|-x,  y+l,  z;  l-x,  l-y,  z; 
2  — z,  2— X,  y;  z+2,  2  — X,  yj 
y+l,  l-z,  x;  l-y,  z+l,  x; 
|-x,  y,  l-z;  l-x,  y,  z+l; 
l-z,  X,  y+l;  z+l,  X,  |-y; 
y+l,  z,  l-x;  l-y,  z,  |-x; 
X,  y+l,  l-z;  X,  l-y,  z+|; 
z,  2— X,  y+2;  z,  2^- X,  ^— y; 
y,  l-z,  l-x;  y,  z+l,  l-x; 
z+i;    x+i  l-y,  z+l; 

x+i  y+i  j-z; 
y+l;    z+l,  x+l,  l-y; 

l-z,  x+l,  y+l; 
l-x;    l-y,  z+l,  x+l; 

y+l,  l-z,  x+l; 
z+l;    x+l,  l-y,  z+l; 

x+l,  y+|.  l-z; 


126  THE   CUBIC   SPACE-GROUPS   T^-Th. 

Space-Group  T^  {continued). 


4     z,   4     X,   4     y 


4~y,  4    z,  4— X 


4    X,  4    y?  4    z 


4     z,  4     X,  4     y 


4~y,  4    z,  4— X 


i-x,  |-y,  i-z 
4~~z,  4— X,  4— y 


4     y,  4     z,  4     X 


Space-Group  T^. 

Two  equivalent  positions: 

(a)  2a. 

Six  equivalent  positions : 

(b)  6e. 

Eight  equivalent  positions : 

(c)  8e. 

Twelve  equivalent  positions : 

(d)  12a.  (e)  12b. 
Sixteen  equivalent  positions : 

(f)  16d. 

Twenty-four  equivalent  positions ; 

(g)  24d. 

Forty-eight  equivalent  positions : 


z+i 

f-x, 

y+i 

z+i  x+f,  i-y; 

f-z,  x-\-l  y-hl; 

y+i 

z+i 

i-x, 

f-y,  z-f-f,  x+i; 

y+i  f-z,  x+i; 

4        X, 

y+i 

z+f 

x+i  i-y,  z+f; 

x+i  y+i  f-z; 

Z  +  i 

l-x, 

y+f 

z+i  x+i  f-y; 

f-z,  x+i  y+f; 

y+i 

z+i 

f-x 

f-y,  z+i  x+f; 

y+i  f-z,  x+f; 

i-x, 

y+i 

Z  +  I 

x+i  f-y,  z+f; 

x+i  y+i  f-z; 

z+i 

l-x, 

y+f 

z+i  x+f,  f-y; 

l-z,  x+i  y+f; 

y+i 

z+i 

f-x 

;     l-y,  z+i  x+f; 

y+i  f-z,  x+i 

xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

x+i  y+i  Z+I;    x+i  i-y,  l-z; 
z+i  x+i  y+i;    |-z,  x+i  ^-y; 


2~x,  y+^,  2  — z; 

?-x,  i-y,  Z+I; 
l-z,  |-x,  y+l; 

z+i  |  — X,  2~yi 


THE   CUBIC  SPACE-GROUPS  T^-TI,. 


127 


Space-Group  T^  {continued). 

y+i  z+i  x+l;    |-y,  |-z,  x+|; 

2~x,  2~y>  2~z;    -J— X,  y+^,  z+2J 

2~Z,     2"~X,     2~y>       2  +  2,     2~X,    y+2J 

i~y>  2~z,  2~x;    y+2>  z+2,  2~x; 

Space-Group  TJ. 

Fowr  equivalent  positions: 

(a)  4b.  (b)  4c. 

Eight  equivalent  positions : 

(c)  8h. 
Twenty-four  equivalent  positions: 


y+2>  2~z,  5— x; 

h-y,  z+i  §-x; 
x+i  |-y,  z+l; 

x+i  y+h  l-z; 
z+i  x+i  l-y; 

2~Z,    X+2>    y4"2J 

l-y,  z+i  x+2; 

y"r2>   I~Z,    X-|-2. 


(d)  xyz;  x+i  |-y,  z 

zxy;  z,  x+i  ^-y 

yzx;  |-y,  z,  x-|-^ 

xyz;  l-x,  y+i  z 

zxy;  z,  |-x,  y+^ 

yzx;  y-hi  z,  ^-x 


X,  y+2j  2~z; 

2"~Z,    X,    y+2; 

y+2)  2~z,  x; 
X,  2""y>  z+2> 
z+2>  X,  5— y; 
2~y>  z+2,  x; 


2~x,  y,  z+2; 
zt"?>  2  X,  y; 
y>  z+2>  2~"X; 
x+i  y,  ^-z; 
l-z,  x+i  y; 
y,  l-z,  x+^ 


Space-Group  TJ. 

^*^/ii  equivalent  positions: 
(a)  8i.  (b)  8e. 

Sixteen  equivalent  positions: 

(c)  16e. 

Twenty-four  equivalent  positions: 

(d)  24e. 

Forty-eight  equivalent  positions : 

(e)  xyz;  x,  y,  ^-z;  |-x,  y,  z;  x,  |-y,  z; 
zxy;  |-z,  X,  y;  z,  ^-x,  y;  z,  x,  |-y; 
yzx;  y,  ^-z,  x;  y,  z,  |-x;  ^-y,  z,  x; 
xyz;  X,  y,  z+|;  x+|,  y,  z;  x,  y+i  z; 
zxy;  z+l,  X,  y;  z,  x+|,  y;  z,  x,  y+§; 
yzx;  y,  z+i  x;  y,  z,  x+^;  y+|,  z,  x; 
x+i  y+h  z+^;  x+i  l-y,  z;  x,  y+|,  |-z; 

|-x.  y,  z-M; 
z+i  x-f-l,  y+l;  z,  x+i  l-y;  ^-z,  x,  y+|; 

z+i,  |-x,  y; 
y+i  z+i  x+l;  |-y,  z,  x+|;  y-H|,  ^-z,  x; 

y,  z-fi  |-x; 


128  THE   CUBIC   SPACE-GROUPS  tJ-T^. 

Space-Group  T^  {continued), 

i~x,  2~y>  2~z;    2— X,  y+27  z;    x,  2~y>  z-j-^; 

x+i  y,  ^-z; 
i-z,  |-x,  |-y;    z,  J-x,  y+|;     z+i  x,  ^-y; 

^-z,  x+l,  y; 
h-y,  i-z,  |-x;    y+i  z,  |-x;    ^-y,  z+J,  x; 

y,  i-z,  x+i 

C.  HEMIMORPHIC  HEMIHEDRY. 
Space-Group  T^. 

One  equivalent  position : 

(a)  la.  (b)  lb. 

Three  equivalent  positions: 
(c)  3a.  (d)  3b. 

Four  equivalent  positions: 

(e)  4a. 

Six  equivalent  positions : 

(f)  6a.  (g)  6d. 
Twelve  equivalent  positions: 

(h)  12f.  (i)  12g. 

Twenty-four  equivalent  positions: 

(j)  xyz; 
zxy; 
yzx; 
yxz; 
xzy; 
zyx; 

Space-Group  T^. 

Four  equivalent  positions: 

(a)  4b.  (c)  4d. 

(b)  4c.  (d)  4e. 

Sixteen  equivalent  positions: 

(e)  16a. 
Twenty-four  equivalent  positions: 

(f)  24a.  (g)  24b. 
Forty-eight  equivalent  positions: 

(h)  48d. 


xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

zyx; 

zyx; 

zyx; 

THE   CUBIC  SPACE-GROUPS  T^-tS. 


129 


Space-Group  T^  (continued). 

Ninety-six  equivalent  positions: 

(i)    xyz;        xyz; 

xyz;        xyz 

zxy;        zxy; 

zxy;        zxy 

yzx;        yzx; 

yzx;        yzx 

yxz;        yxz; 

yxz;        yxz 

xzy;        xzy; 

xzy;        xzy 

zyx;        zyx; 

zjrx;         zyx 

x+i  y+i  z; 

x+i  h-y,  z; 

l-x,  y+i  z 

l-x,  l-y,  z 

z+i  x+i  y; 

^-z,  x+i  y; 

l-z,  l-x,  y 

z+i  l-x,  y 

y+i  z+l,  x; 

l-y,  1-z,  x; 

y+i  l-z,  X 

l-y,  z+i  X 

y+i  x+i  z; 

i-y,  x+i  z; 

y+i  l-x,  z 

;   l-y,  l-x,  z 

x+i  z+l,  y; 

x+i  l-z,  y; 

l-x,  l-z,  y 

;    l-x,  z+i  y 

z+i  y+i,  x; 

l-z,  l-y,  x; 

l-z,  y+i  X 

;    z+i  l-y,  X 

x+i  y,  z+l; 

x+iy,  ^-z; 

l-x,  y,  l-z 

l-x,  y,  Z+I 

z+i,x,  y+l; 

i-z,  X,  l-y; 

l-z,  X,  y+l 

z+i  X,  l-y 

y+i  z,  x+i; 

l-y,  z.  x+i; 

y+i  z,  l-x 

;    l-y,  z,  l-x 

y+i  X,  z+^; 

i-y,  X,  i-z; 

y+i  X,  l-z 

l-y,  X,  Z+I 

x+i  z,  y+l; 

x+i  z,  i-y; 

l-x,  z,  y+l 

l-x,  z,  l-y 

z+i  y,  x+^; 

l-z,  y,  X+I; 

l-z,  y,  l-x 

z+i  y,  l-x 

X,  y+i  z+l; 

X,  i-y,  l-z; 

X,  y+i  l-z 

X,  l-y,  Z+I 

z,  x+i  y+l; 

z,  x+i  l-y; 

z,  l-x,  y+l 

z,  l-x,  l-y 

y,  z+i  x+^; 

y,  l-z,  x+i 

y,  l-z,  l-x 

y,  z+i  l-x 

y,  x+i  z-\-i; 

y,  x+i  i-z; 

y,  l-x,  l-z 

y.  l-x,  Z+I 

X,  z+l,  y+^; 

X,  l-z,  i-y; 

X,  l-z,  y+l, 

X,  z+i  l-y 

z,  y+i  x+l; 

z,  l-y,  X+I; 

z,  y+i  l-x 

z,  l-y,  l-x 

Space-Group  T|. 

Two  equivalent  positions: 

(a)  2a. 

Six  equivalent  positions: 

(b)  6e. 

Eight  equivalent  positions: 

(c)  8a. 

Twelve  equivalent  positions: 

(d)  12h.  (e)  12a. 
Twenty-four  equivalent  positions: 

(f)  24f.  (g)  24g. 

Forty-eight  equivalent  positions: 


xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

yxz; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

xzy; 

zyx; 

^; 

zyx; 

zyx; 

130 


THE    CUBIC   SPACE-GROUPS   T?-!^. 


Space-Group  T^  {continued). 
x+i  y+l,  z+l 

z+i  x+l,  y+l 

y+i  z+i  x+l 

y+i,  x+l,  z+l 

x+i  z+l,  y+l 

z+i  y+i  x+i 


x+2)  ^~yj  2~z; 

^— z,  x+2,  2~~yi 

i-y,  l-z,  x+l; 
2~y>  x+2,  2~~z; 
x+i  |-z,  J-y; 
§-z,  l-y,  x+l; 


Space-Group  T\. 

Two  equivalent  positions: 

(a)  2a. 

Six  equivalent  positions: 

(b)  6e.  (d)  6g. 

(c)  6f. 

Eight  equivalent  positions: 

(e)  8a. 

Twelve  equivalent  positions : 

(f)  12a.  (h)  12j. 

(g)  12i. 

Twenty-four  equivalent  positions : 

(i)  xyz;  xyz;  xyz;  xyz; 
ixy;  zxy;  zxy;  zxy; 
yzx;        yzx;        yzx;        yzx; 

y+l,  x+l,  z+\;    i-y,  x-f-|,  |-z; 

x+i  z+i  y+l;    x+i  l-z,  |-y; 

z+i  y+i  x+l;    l-z,  |-y,  x+|; 

Space-Group  T^. 

Eight  equivalent  positions: 

(a)  8i.  (b)  Be. 

Twenty-four  equivalent  positions: 

(c)  24c.  (d)  24h. 

Thirty-two  equivalent  positions : 

(e)  32c. 


2~x,  y+2j  2~z; 

2~x,  2~y>  Z+2I 

2     z,  2     X,  y+2J 

Z+2>     2~X,     2~yj 

y'T2>  2~z,  2"~x; 

2"~y>  z+2,  ^~x; 
yi2>   2~x,  2~z; 

^~y,  2~x,  z+2; 
2~"X,  ^  — z,  y+2j 

^  — X,  z+2,  2~y; 
§-z,  y+i  l-x; 

z+2>  2~y>  2~~x; 


y+i  ¥-x,  l-z; 

5~yj  5~x,  z+2; 
2~x,  ^  — z,  y+^; 

|-x,  z+l,  |-y; 
i~z,  y+2)  ?~~x; 

z+l,  l-y,  l-x. 


THE   CUBIC   SPACE-GROUPS   T^-I^.  131 

Space-Group  T*  (continued). 

Forty-eight  equivalent  positions: 

(f)  48e.  (g)  48a. 

Ninety -six  equivalent  positions: 

(h)  xyz;  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy;  zxy; 
yzx;  yzx;  yzx;  yzx; 
y+i  x+i  z+i;    |-y,  x+^,  |-z;    y+i  ^-x,  §-z; 

h-y,  i-x,  z+^; 
x+i  z+l,  y+l;    x+i  ^-z,  |-y;    ^-x,  |-z,  y+l; 

2~X,    Z+2>     2~y> 

z+i  y+i  x+^;     ^-z,  |-y,  x+|;     |-z,  y+i  ^-x; 

z+i  h-y,  ^-x; 
x+i,  y+iz;    x+i|-y,  z;     |-x,  y+i  z;     ^-x,  |-y,  z; 
z+l,  x+l,  y;    l-z,  x+l,  y;     5-z,  |-x,  y;     z+i  ^-x,  y; 
y+i  z+i  x;     |-y,  i-z,  x;    y+i  |-z,  x;     |-y,  z+i  x; 


y,  X,  z-h^; 

y,  X,  ^-z; 

y,  X, 

l-z; 

y,  X,  z+^; 

X,  z,  y+l; 

X,  z,  |-y; 

X,  z, 

y+l; 

X,  z,  l-y; 

z,  y,  x+^; 

z,  y,  x+l; 

z,  y, 

i  — V 
2        ■''■> 

z,  y,  5-x; 

x+l,  y,  z+^;  x+iy  |-z;  |-x,  y,  ^-z;  |-x,  y,  z+|; 

z+ix,  y-l-^;  ^-z,  x,  |-y;  *-z,  x,  y+l;  z+|,  x  |-y; 

y+i,  z,  x+l;  i-y,  z,  x+l;  y+|,  z,  ^-x;  |-y,  z,  ^-x; 

y,  x+i  z;  y,  x+l,  z;  y,  |-x,  z;  y,  |-x,  z; 

X,  z+i  y;  X,  l-z,  y;  x,  |-z,  y;  x,  z+|,  y; 

z,  y+l,  x;  z,  |-y,  x;  z,  y+i  x;  z,  |-y,  x; 

X,  y+iz+l;  X,  |-y,  |-z;  x,  y+i  ^-z;  x,  J-y,  z+|; 

z,  x+iy+l;  z,  x+l,  §-y;  z,  |-x,  y+|;  z,  |-x,  |-y; 

y,  z+l,  x+l;  y,  l-z,  x+l;  y,  |-z,  |-x;  y,  z+|,  |-x; 

y+l,  X,  z;  |-y,  x,  z;  y+|,  x,  z;  |-y,  x,  z; 

x+l,  z,  y;  x+l,  z,  y;  |-x,  z,  y;  |-x,  z,  y; 

z+i  y,  x;  |-z,  y,  x;  |-z,  y,  x;  z+|,  y,  x. 

Space-Group  T^. 

Twelve  equivalent  positions : 

(a)  12k.  (b)  121. 

Sixteen  equivalent  positions : 

(c)  16f. 

Twenty-four  equivalent  positions : 

(d)  24i. 

Forty-eight  equivalent  positions : 

(e)  xyz;  x,  y,  |-z;  |-x,  y,  z;  x,  |-y,  z; 
zxy;  |-z,  X,  y;  z,  |-x,  y;  z,  x,  |-y; 
yzx;    y,  |-z,  x;    y,  z,  |-x;    |-y,  z,  x; 


132  THE   CUBIC  SPACE-GROUPS  Tfl-O^ 

Space-Group  T'  {continued). 

y+i  x+l,  z+i;     i-y,  x+\,  f-z;     y+i  f-x,  |-z; 

4~y>    4~X,    Z  +  jI 

x+i  z+l,  y+l;     x+i  f-z,  \-y;     f-x,  f-z,  y+f; 

4— X,  z+j,  4~yj 

z+i,  y+i  x+f;     f-z,  f-y,  x+f;     f-z,  y+f,  f-x; 

z+4>  4~y>  4~x; 
x+J,y+|,z+|;       x+l,  ^-y,  z;  x,  y+|,  |-z; 

l-x,  y,  z+^; 
z,  x+l,  ^-y;  i-z,  X,  y+|; 

z+2)  2~x,  y; 
|-y,  z,  x+l;  y+i,  |-z,  x; 

y,  z+l,  ^-x; 
l-y,  x-hf,  f-z;    y+f,  f-x,  f-z; 

4~y>  4~x,  z-f-i) 

x+f,  f-z,  f-y;    f-x,  f-z,  y+f; 

4~x,  z+4,  4— y; 


z+i  x+l,  y+l 
y+i  z+i  x+l 
y+i  x+f,  z+f 
x+f,  z+f,  y+f 
z+f,  y+f,  x+f 


■z,  4,    y,  x+4;    4— z,  y+4,  4~x; 

z+f,  f-y,f-x. 


D.  ENANTIOMORPHIC  HEMIHEDRY. 
Space-Group  O*. 

One  equivalent  position: 

(a)  la.  (b)  lb. 

Three  equivalent  positions: 
(c)  3a.  (d)  3b. 

Six  equivalent  positions: 

(e)  6a.  (g)  6c. 

(f)  6b.  (h)  6d. 

Eight  equivalent  positions: 

(i)  8c. 
Twelve  equivalent  positions: 

G)  12m.  (k)  12n. 

Twenty-four  equivalent  positions: 

(1)      xyz;  xyz;  xyz; 

zxy;  zxy;  zxy; 

yzx;  yzx;  yzx; 

yxz;  yxz;  yxz; 

xzy;  xzy;  xzy; 

zyx;  zyx;  zyx; 


xyz; 
zxy; 
yzx; 
yxz; 
xzy; 
zyx. 


THE   CUBIC  SPACE-GROUPS  O^-O'.  133 

Space-Group  O^. 

Two  equivalent  positions: 

(a)  2a. 

Four  equivalent  positions: 

(b)  4d.  (c)  4e. 
Six  equivalent  positions: 

(d)  6e.  (f)   6g. 

(e)  6f. 

Eight  equivalent  positions: 

(g)  8d. 
Twelve  equivalent  positions: 

(h)  12a.  (k)  12o. 

(i)  12i.  (1)  12p. 

a)  12j. 
Twenty-four  equivalent  positions: 

(m)  xyz;  xyz;  xyz;  xyz: 
zxy;  zxy;  zjcy;  zxy; 
yzx;  yzx;  yzx;  yzx; 
h-y>  i-x,  i-z;    y+i  |-x,  z+|;    ^-y,  x+J,  z+|; 

y+i  x+i  i-z; 
^-x,  i-z,  ^-y;    i-x,  z+i  y+|;    x+|,  z+|,  ^-y; 

x+l,  i-z,  y+l; 
i-z,  ^-y,  l-x;    z-hi  y+h  ^-x;    z+|,  ^-y,  x+J; 

^-z,  y+l,  x+^. 

Space-Group  0'. 

Four  equivalent  positions: 

(a)  4b.  (b)  4c. 

Eight  equivalent  positions: 

(c)  8e. 

Twenty-four  equivalent  positions : 

(d)  24c.  (e)  24a. 
Thirty-two  equivalent  positions: 

(f)  32a. 

Forty-eight  equivalent  positions : 

(g)  48f.  (i)  48a. 
(h)  48g. 


134 


THE   CUBIC   SPACE-GROUPS  0^-0*. 


Space-Group  0^  {continued). 

Ninety-six  equivalent  positions: 


(j)  xyz;        xyz; 

xyz;        xyz 

zxy;        zxy; 

zxy;        zxy 

yzx;        yzx; 

yzx;        yzx 

yxz;        yxz; 

yxz;        yxz 

xzy;        xzy; 

xzy;        xzy 

zyx;        zyx; 

zyx;        zyx 

x+i  y+i  z 

x+l,  i-y,  z 

i-x,  y+i  z 

l-x,  |-y,  z; 

z+i  x+l,  y 

;    ^-z,  x+i  y 

A-z,  i-x,  y 

r    z+l,  l-x,  y; 

y+i  z+i  X 

i-y,  ^-z,  X 

y+i  5-z,  X 

l-y,  z+l,  x; 

^-y,  l-x,  z 

y+i  l-x,  z 

l-y,  x+i  z 

y+l,  x+l,  z; 

i-x,  l-z,  y 

l-x,  z+i  y 

x+l,  z+i  y 

x+l,  l-z,  y; 

l-z,  ^-y,  X 

,     z+i  y+l,  X 

z+i  l-y,  X 

l-z,  y+l,  x; 

x+i  y,  z+^ 

x+i  y,  i-Z; 

l-x,  y,  l-z 

l-x,  y,  z+l; 

z+i  X,  y-l-^ 

i-z,  X,  ^-y 

l-z,  X,  y+l 

z+l,  X,  l-y; 

y+i  z,  x+^ 

^-y,  z,  x+l 

y+i  z,  |-x 

l-y,  z,  l-x; 

^-y,  X,  l-z 

y+l,  X,  z+l 

h-Y,  X,  z+l 

y+l,  X,  l-z; 

i-x,  z,  l-y 

l-x,  z,  y+^; 

x+i  z,  |-y 

x+l,  z,  y+l; 

|-z,  y,  ^-x 

z+i  y,  l-x 

z+l,  y,  x+l 

l-z,  y,  x+l; 

X,  y+l,  z+l 

X,  ^-y,  l-z, 

X,  y+l,  l-z 

X,  l-y,  z+l; 

z,  xH-l,  y+l 

z,  x+i  |-y 

z,  l-x,  y+l 

z,  l-x,  l-y; 

y,  z+i  x+l 

y,  i-z,  x+h 

y,  l-z,  l-x 

y,  z+l,  l-x; 

y,  l-x,  l-z 

y,  ^-x,  z+l 

y,  x+l,  z+l 

y,  x+l,  l-z; 

X,  i-z,  l-y 

X,  z+i  y+l, 

X,  z+l,  |-y, 

X,  l-z,  y+l; 

z,  ^-y,  i-x 

z,  y+i  i-x, 

z.  l-y,  x+l 

z.  y+l,  x+|. 

Space-Group  0^. 

Eight  equivalent  positions: 

(a)  8f.  (b)  8g. 

Sixteen  equivalent  positions: 

(c)  16b.  (d)  16c. 

Thirty-two  equivalent  positions: 

(e)  32b. 

Forty-eight  equivalent  positions: 

(f)  48c.  (g)  48h. 
Ninety-six  equivalent  positions: 

(h)  xyz;  xyz;  xyz; 
zxy;  zxy;  zjiy; 
yzx;        yzx;        yzx; 


xyz; 
zxy; 
yzx; 


i-y,  i- 


X,  i-z;    y+i  \- 


•X,  i-z,  i-y;    i-x,  z+l. 


z+i;   l-y,  x+i,  z-\-\) 

y+i,  x+i  i-z; 
y+i;    x+i,  z+i  i-y; 

x+i  i-z,  y+J; 


THE   CUBIC   SPACE-GROUPS   0*-0^. 


135 


Space-Group  O*  (continued). 


—  z. 


y,  l-x;    z+i  y+i  ^-x;     z+i  i-y,  x-\-i; 

i-z,  y-l-i,  x+i; 
x+iy+iz;    x-l-i^-y,  z;     |-x,  y+i  z;     |-x,  |-y,  z; 
z+ix-|-|,  y;    i-z,  x-j-^  y;     |-z,  ^-x,  y;     z+i  |-x,  y; 
y+iz+l,  x;     ^-y,  i-z,  x;    y+i  |-z,  x;     |-y,  z+^,  x; 
f-y,  f-x,  i-z;    y+l  f-x,  z+i;    f-y,  x+f,  z+i; 

y+i  x+i  i-z; 
f-x,  f-z,  i-y;    f-x,  z+f,  y+i;    x+f,  z+f,  f-y; 

x+f,  f-z,  y+i; 
f-z,  f-y,  i-x;     z+f,  y+f,  i-x;     z+f,  f-y,  x+i; 

?  — z,  y+4,  x+j; 


x+iy,  z+l;    x+iy,  i-z; 


2  ~  X,  y,  2  —  z ; 


•X,  y,  z+^; 


z+ix,  y+^;  |-z,  x,  |-y;  ^-z,  x,  y+^;  z+i  x,  |-y; 
y+iz,  x+l;  ^-y,  z,  x+^;  y+i  z,  |-x;  ^-y,  z.  §-x; 
f-y,  ¥-x,  f-z;    y+f,  i-x,  z+f;     f-y,  x+f,  z+f; 

y+f,  x+f,  f-z; 
f-x,  f-z,  f-y;    f-x,  z+f,  y+f;    x+f,  z+f,  f-y; 

x+f,  l-z,  y+f; 
f-z,  f-y,  f-x;    z+f,  y+f,  f-x;    z+f,  f-y,  x+f; 

f-z,  y+f,  x+f; 
X,  y+iz+^;    x,  ^-y,  ^-z;    x,  y+|,  ^-z;     x,  i-y,  z+|; 
z,  x+iy+l;    z,  x+i|-y;    z,  ^-x,  y+|;     z,  |-x,  |-y; 
y,  z+ix+^;    y,  |-z,  x+l;    y,  ^-z,  |-x;    y,  z+i  |-x; 
-y,  f-x,  f-z;    y+f,  f-x,  z+f;    f-y,  x+f,  z+f; 

y+i  x+f,  f-z; 


■X,  f-z,  f-y; 


i_ 


X,  z+f,  y+f;    x+f,  z+f,  f-y; 

x+f,  f-z,  y+f; 


-z,  f-y,  f-x;     z+f,  y+f,  f-x;     z+f,  f-y,  x+f; 


f-z,  y+f,  x+f 


Space-Group  0^. 

Two  equivalent  positions: 

(a)  2a. 

Six  equivalent  positions: 

(b)  6e. 

Eight  equivalent  positions: 

(c)  8e. 

Twelve  equivalent  positions: 

(d)  12h.  (f)  12b. 

(e)  12a. 

Sixteen  equivalent  positions: 
(g)  16d. 


136 


THE   CUBIC  SPACE-GEOUPS  0*-0'*. 


Space-Group  0^  (continued). 

Twenty-four  equivalent  positions: 

(h)  24j.  (i)  24k. 

Forty-eight  equivalent  positions: 


xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

yxz; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

xzy; 

zyx; 

zyx; 

zyx; 

zyx; 

x+i  y+h  z+§;    x+i  ^-y,  i-z; 


z+i  x+i  y+J;    5-z,  x+i  |-y; 


^— X,  y+^,  ^  — z; 

i-x,  h-y,  z+^; 
2~z,  ^— X,  y+a; 
z+i  l-x,  ^-y; 
y+i  z+i  x+l;    l-y,  l-z,  x+l;    y+J,  |-z,  |-x; 

i-y,  z+i  ^-x; 
l-y,  l-x,  |-z;    y+i  §-x,  z+l;    |-y,  x+|,  z+i; 

y+i  x+i  |-z; 
i-x,  i-z,  |-y;    l-x,  z+i  y+|;    x+|,  z+|,  ^-y; 

x+l,  i-z,  y+l; 
i-z,  |-y,  |-x;    z+i  y+l,  ^-x;    z+i  |-y,  x-f|; 

l-z,  y+5,  x+i. 
Space-Group  0". 

Four  equivalent  positions : 

(a)  4g.  (b)  4h. 

Etgf/i<  equivalent  positions: 

(c)  8j. 

Twelve  equivalent  positions: 

(d)  12q. 

Twenty-four  equivalent  positions: 

(e)  xyz;    x-fl,  |-y,  z;    x,  y+i  |-z;    |-x,  y,  z-|-|; 


zxy;    z,  x-ff,  §-y 


z,  X,  y+l;    z+i  i-x,  y; 


yzx;    ^-y,  z,  x-j-|;    y-f-i  |-z,  x;    y,  z+|,  §-x; 
i-y,  i-x,  i-z;    y+i  |-x,  z+i;    f-y,  x+i  z+f; 

y+J,  x-l-f,  f-z; 
i-x,  i-z,  J-y;    l-x,  z-fi  y+|;    x+i  z+f,  |-y; 

x+f,  f-z,  y+i; 
l-z,  f-y,  f-x;    z+f,  y+f,  f-x;    z+f,  f-y,  x+f; 

f-z,  y+f,  x+f. 

It  is  evident  that  a  suitable  transformation  would  simplify  the  two  unique 
cases. 


THE   CUBIC   SPACE-GROUPS  O'^-O*.  I'M 

Space-Group  O^. 

Four  equivalent  positions: 

(a)  4i.  (b)  4j. 

Eight  equivalent  positions: 

(c)  8k. 

Twelve  equivalent  positions: 

(d)  12r. 

Twenty-four  equivalent  positions: 

(e)  xyz;  x+i  ^-y,  z;  x,  y-j-|,  |-z;  |-x,  y,  z+|; 
zxy;  z,  x+i  ^-y;  ^-z,  x,  y+|;  z+|,  |-x,  y; 
yzx;  |-y,  z,  x+|;  y+i  ^-z,  x;  y,  z+i  ^-x; 
l-y,  f-x,  l-z;     y+i,  i-x,  z+f;  i-y,  x+f,  z+i; 

y+4,  x-f-j,  4  — z; 
l-x,  f-z,  l-y;     i-x,  z+f,  y+i;     x+f,  z+i,  |-y; 

x+i  i-z,  y+l; 
l-z,  l-y,  l-x;     z+f,  y+i  i-x;     z+i  l-y,  z+f; 

i-z,  y+f,  x+i 

Space-Group  O*. 

Eight  equivalent  positions : 

(a)  81.  (b)  8m. 

Twelve  equivalent  positions: 

(c)  12s.  (d)  121. 

Sixteen  equivalent  positions : 

(e)  16g. 

Twenty-four  equivalent  positions : 

(f)  241.  (g)  24m.  (h)  24n. 

Forty-eight  equivalent  positions: 

(i)  xyz;  x,  y,  i-z;  ^-x,  y,  z;  x,  ^-y,  z; 
zxy;  i-z,  X,  y;  z,  ^-x,  y;  z,  x,  ^-y; 
yzx;  y,  |-z,  x;  y,  z,  |-x;  ^-y,  z,  x; 
¥-y,  i-x,  i-z;     y+i  i-x,  z+f;     f-y,  x+f,  z+i; 

y"!"*,  x+4,  4  — z; 
i-x,  i-z,  l-y;     i-x,  z+f,  y+i;     x+f,  z+f,  f-y; 

x+f,  f-z,  y+f; 
f-z,  f-y,  f-x;     z+f,  y+f,  f-x;     z+f,  f-y,  x+f; 

f-z,  y+l,  x+f; 
x+i  y+l,  z+l;     x+i  l-y,  z;     x,  y+i  |-z; 

l-x,  y,  z+l; 


138  THE   CUBIC  SPACE-GROUPS   0*-0i. 

Space-Group  O*  (continued). 

z+i,  x+i  y+l;    z,  x+l,  ^-y;    |-z,  x,  y+^; 

z+l,  l-x,  y; 
y+i  z+i  5C+I;     l-y,  z,  x+|;     y+i  l-z,  x; 

y,  z+i  l-x; 
f-y,  l-x,  f-z;     y+l,  f-x,  z+i;     f-y,  x+i  z+f; 

y"r4)  x-|-4,  4  — z; 
f-x,  f-z,  l-y;     f-x,  z+i  y+f;     x+i  z+f,  f-y; 

x+f,  f-z,  y+f; 
l-z,  f-y,  f-x;     z+f,  y+f,  f-x;     z+f,  f-y,  x+f; 

f-z,  y+f,  x+f. 

E.  HOLOHEDRY. 
Space-Group  OJ. 

One  equivalent  position: 

(a)  la.  (b)  lb. 

Three  equivalent  positions: 

(c)  3a.  (d)  3b. 

Six  equivalent  positions: 

(e)  6a.  (f)   6d. 

Eight  equivalent  positions: 

(g)  8c. 

Twelve  equivalent  positions: 

(h)  12f.  (j)  12n. 

(i)    12m. 

Twenty-four  equivalent  positions: 

(k)  24o.  (m)  24q. 

a)    24p. 

Forty-eight  equivalent  positions: 

(n)    xyz;  xyz;  xyz:  xyz; 

zxy;  zxy;  zxy;  zxy; 

yzx;  yzx;  yzx;  yzx; 

yxz;  yxz;  yxz;  yxz; 

xzy;  xzy;  xzy;  xzy; 

zyx;  zyx;  zyx;  zyx; 

xyz;  xyz;  xyz;  xyz; 

zxy;  zxy;  zxy;  zxy; 

yzx;  yzx;  yzx;  yzx; 

yxz;  yxz;  yxz;  yxz; 

xzy;  xzy;  xzy;  xzy; 

zyx;  zyx;  gyx;  zyx; 


THE   CUBIC   SPACE-GROUPS   0,-0^.  139 

Space  Group  01. 

Two  equivalent  positions : 

(a)  2a. 

<Sta;  equivalent  positions: 

(b)  6e. 

Eight  equivalent  positions: 

(c)  8e. 

Twelve  equivalent  positions : 

(d)  12h.  (e)  12a. 
Sixteen  equivalent  positions: 

(f)  16d. 

Twenty-four  equivalent  positions : 

(g)  24f.  (h)24j. 
Forty-eight  equivalent  positions: 

(i)  xyz;  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy;  zxy; 
yzx;  yzx;  yzx;  yzx; 
yxz;  yxz;  yxz;  yxz; 
xzy;  xzy;  xzy;  xzy; 
z^x;  zyx;  zyx;  zyx; 
i-x,  ^-y,  |-z;    l-x,  y+i  z+^;    x+i  i-y,  z+l; 

x+i  y+h  i-z; 
\-z,  i-x,  |-y;     z+§,  |-x,  y+i;     z+i  x+|,  ^-y; 

^— z,  x+2>  y+ij 

2~y>  2~z,  2~x;    y+f,  z+2,  2~x;     2~y>  z+2,  x+2> 

y~r2j  2  z,  X+2J 
y+i  x+i  z+i;    \-y,  x+l,  §-z;    y+i  |-x,  \-z; 

l~y,    2~X,    z+2; 

x+i  z+i  y+^;    x+l,  §-z,  ^-y;    §-x,  |-z,  y+^; 

l-x,  z+l,  |-y; 
z+i  y+i  x+l;     l-z,  |-y,  x+^;    |-z,  y+|,  |-x; 

z+i  5-y;  i-x. 
Space-Group  0^. 

Two  equivalent  positions: 

(a)  2a. 

Six  equivalent  positions: 

(b)  6e.  (c)  6f.  (d)  6g. 
Eight  equivalent  positions : 

(e)  8e. 


140  THE   CUBIC   SPACE-GROUPS   0^-0^. 

Space-Group  0^  (continued). 
Twelve  equivalent  positions: 

(f)  12a.  (g)  12i.  (h)  12j. 

Sixteen  equivalent  positions: 

(i)  16d. 
Twenty-four  equivalent  positions: 

(j)  24s.  (k)  24r. 

Forty-eight  equivalent  positions : 

(1)  xyz;  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy;  zxy; 
yzx;  yzx;  yzx;  yzx; 
^-y,  l-x,  l-z;    y+i,  i-x,  z+i;     ^-y,  x+i  z+|; 

y~r2>   ^+2,    2~z; 
l-x,  ^-z,  l-y;     l-x,  z+l,  y+|;     x+i  z+|,  |-y; 

X+2J  2~z,  y+2j 
h-z,  l-y,  l-x;     z+i  y+i,  i-x;    z+i  |-y,  x+|; 

2~z,  y+2,  X+2J 
xyz;        xyz;        xyz;        xyz; 
zxy;        zxy;        zxy;        zxy; 
ygX;        yzx;        yzx;        yzx; 
y+i,  x+l,  z+l;    l-y,  x+|,  |-z;    y+i  |-x,  i-z; 

l-y,  l-x,  z+l; 
x+l,  z+l,  y+l;     x+l,  l-z,  |-y;     |-x,  |-z,  y+|; 

2— X,  z+2,  ^~y; 
z+l,  y+l,  x+l;    l-z,  l-y,  x+l;    |-z,  y+|,  |-x; 

z+l,  l-y,  l-x. 

Space-Group  0^. 

Two  equivalent  positions: 

(a)  2a. 

Four  equivalent  positions: 

(b)  4d.  (c)  4e. 
Six  equivalent  positions : 

(d)  6e. 

Eight  equivalent  positions : 

(e)  8d. 

Twelve  equivalent  positions : 

(f)  12h.  (g)  12a. 
Twenty-four  equivalent  positions: 

(h)  24f.  (i)  24t.  (j)  24u. 


THE   CUBIC   SPACE-GROUPS   Oh-Oh. 


141 


Space-Group  O^  (continued). 

Forty-eight  equivalent  positions: 

(k)  xyz;  xyz;  xyz;  xyz; 
zxy;  zxy;  zxy;  zxy; 
yzx;  yzx;  yzx;  yzx; 
a—y?  2~x,  2~z;     y-r2)  2~x,  z+2 

A-x,  l-z,  |-y;    ^-x,  z+l,  y+| 

i-z,  ^-y,  i-x;    z+i  y+i  ^-x 

^-x,  ^-y,  i-z;    |-x,  y+i  z+^ 

2"~Z,     2~X,     2~y>       ^12)     2~X,    y+2 

|-y,  ^-z,  ^-x;    y+i  z+^,  |-x 


yxz; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

xzy; 

zyx; 

zyx; 

zyx; 

zyx. 

Space-Group  0^. 

Four  equivalent  positions: 

(a)  4b.  (b)  4c. 

^zfif/i^  equivalent  positions: 

(c)  8e. 

Twenty-four  equivalent  positions: 

(d)  24c.  (e)  24a. 
Thirty-two  equivalent  positions: 

(f)  32a. 

Forty-eight  equivalent  positions: 

(g)  48a.  (h)  48f.  (i)  48g. 
Ninety-six  equivalent  positions : 

(j)   96a.  (k)  96b. 

One  hundred  ninety-two  equivalent  positions: 
(1)     xyz;        xyz;        xyz;        xyz; 


zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

yxz; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

xzy; 

zyx; 

zyx; 

zyx; 

zyx; 

xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

2~y<  xH-2,  Z+2I 

y+i  x-l-l,  |-z; 
x+2)  Z+2,  ^— y; 

X+2>  2~z,  y+2> 

z+2,  2~yj  X+2J 
l-z,  y+i  x+l; 

x+i  5-y,  z+l; 
x+l,  y+i  l-z; 

z+2,  X+2,  2~yj 

2  — Z,  X+2,  y+2  • 

2~y>  z+2,  x+2; 

y+2,  |-z,  x+^; 


142 


THE   CUBIC  SPACE-GROUPS  O^-OS. 


Space-Group  0^  (continued). 


yzx;        yzx; 

yzx;        yzx; 

yxz;        yxz; 

yxz;        yxz; 

xzy;        xzy; 

xzy;        xzy; 

zyx;        zyx; 

zyx;        zyx; 

x+l,  y+i  z; 

x+i  ^-y,  z; 

l-x,  y+l,  z; 

2    X,  2    y,  z 

z+i  x+i  y; 

2~z,  x-i-2>  y> 

l-z,  l-x,  y; 

z+l,  l-x,  y 

y+i  z+i  X, 

h-y,  2-z,  X, 

y+l,  l-z,  X, 

l-y,  z+l,  X 

2~y>  2~x,  z; 

y+h  l-x,  z; 

l-y,  x+l,  z, 

y+l,  x+l,  z 

l-x,  l-z,  y; 

2~x,  z-|-2,  y, 

x+l,  z+l,  y, 

x+l,  l-z,  y 

f-z,  |-y,  x; 

z+l,  y+i  X, 

Z+2J  2~y>  X 

l-z,  y+l,  X 

l-x,  |-y,  z, 

^-x,  y+i  z 

x+l,  |-y,  z 

x+l,  y+l,  z 

l-z,  l-x,  y; 

z+2)  2     X,  y; 

z+l,  x+l,  y; 

l-z,  X+l,  y 

l-y,  i-z,  X, 

y+l,  z+ix; 

l-y,  z+l,  X 

y+l,  l-z,  X 

y+i  x+i  z 

l-y,  x+l,  z 

y+l,  l-x,  z 

l-y,  l-x,  z 

x+l,  z+i  y 

x+i  l-z,  y 

l-x,  l-z,  y 

l-x,  z+l,  y 

z+i  y+i  X 

l-z,  ^-y,  X 

l-z,  y+l,  X 

z+l,  l-y,  X 

x+i  y,  z+i 

x+i  y,  l-z 

l-x,  y,  l-z 

l-x,  y,  z+l 

z+i  X,  y+J 

l-z,  X,  l-y 

l-z,  X,  y+l 

z+l,  X,  l-y 

y+l,  z,  x+^ 

l-y,  z,  x+^ 

y+l,  z,  l-x 

,    l-y,  z,  l-x 

^-y,  X,  i-z 

;    y+i,  X,  z+i 

,    l-y,  X,  z+l 

;   y+l,  X,  |-z 

^-x,  z,  ^-y 

;    |-x,  z,  y+l 

;    x+l,  z,  l-y 

;    x+l,  z,  y+l 

h-2,  y,  2-x 

;     z+l,  y,  l-x 

;     z+l,  y,  x+l 

;    l-z,  y,  x+l 

l-x,  y,  i-z 

,    |-x,  y,  z+l 

x+l,  y,  z+l 

,    x+l,  y,  l-z 

i-z,  X,  l-y 

;     z+l,  X,  y+l 

;    z+l,  X,  l-y 

;    l-z,  X,  y+l 

l-y,  z,  ¥-x 

;   y+i  z,  i-x 

;    l-y,  z,  x+l 

;    y+l,  z,  3^+1 

y+i  X,  z+l 

;   l-y,  X,  |-z 

;   y+l,  X,  l-z 

;    l-y,  X,  z+l 

x+i  z,  y+l 

;    x+i  z,  |-y 

;    |-x,  z,  y+l 

;    l-x,  z,  l-y 

z+l,  y,  x+l 

;   l-z,  y,  x+l 

;    l-z,  y,  l-x 

;    l-z,  y,  l-x 

X,  y+l,  z+l 

;    X,  l-y,  l-z 

;    X,  y+l,  l-z 

;    X,  l-y,  z+l 

z,  x+i  y+l 

;    z,  x+i  |-y 

;    z,  l-x,  y+l 

;    z,  l-x,  l-y 

y,  z+l,  x+l 

,    y,  l-z,  x+l 

;   y,  l-z,  l-x 

;   y,  z+l,  l-x 

y,  l-x,  l-z 

;   y,  l-x,  z+i 

;   y,  x+l,  z+l 

;   y,  x+l,  l-z 

X,  i-z,  i-y 

;    X,  z+i  y+l 

;    X,  z+l,  l-y 

;    X,  l-z,  y+l 

z,  l-y,  §-x 

;     z,  y+i  l-x 

;     z,  l-y,  x+l 

;    z,  y+l,  x+l 

X,  h-J,  h-z 

;    X,  y+l,  z+l 

;    X,  l-y,  z+l 

;    X,  y+l,  l-z 

z,  ^-x,  |-y 

;     z,  l-x,  y+l 

;    z,  x+l,  l-y 

;    z,  x+l,  y+l 

y,  l-z,  l-x 

;   y,  z+l,  l-x 

;    y,  z+l,  x+l 

;    y,  l-z,  x+l 

y,  x+l,  z+l 

;   y,  x+l,  |-z 

;   y,  l-x,  l-z 

;   y,  l-x,  z+l 

X,  z+i  y+l 

;    X,  l-z,  |-y 

;    X,  l-z,  y+l 

;    X,  z+l,  l-y 

z,  y+l,  x+l 

;    z,  l-y,  x+l 

;    z,  y+l,  l-x 

•     z,  l-y,  l-x 

Space-Group  0^. 

Eight  equivalent  positic 

ns: 

(a)  8i.               (b) 

8e. 

Twenty-four  equivalent 

positions : 

(c)  24c.                (d 

)  24h. 

THE   CUBIC   SPACE-GROUP   O^.  143 


Space-Group  0°  (continued). 

Forty-eight  equivalent  positions: 

(e)  48a.  (f)  48e. 

Sixty-four  equivalent  positions : 

(g)  64a. 
Ninety-six  equivalent  positions: 

(h)  96c.  (i)  96d. 

One  hundred  ninety-two  equivalent  positions: 
(J)  xyz;        xyz;        xyz;        xyz; 


zxy;        zxy; 

zxy; 

zxy; 

yzx;        yzx; 

yzx; 

yzx; 

yxz;       yxz; 

yxz; 

yxz; 

xzy;        xzy; 

xzy; 

xzy; 

zyx;         zyx; 

zyx; 

zyx; 

l-x,  l-y,  h- 

-z;    h- 

-X,  y+i  z+l; 

x+l,  ^-y,  z+^; 
x+i  y+i  ^-z; 

2     z,  2     X,  2     y;    z-t-2,  2     X,  y+2;    Z+2,  X+2>  2~y; 

l-z,  x+i  y+i; 
l-y,  ^-z,  l-x;    y+l,  z+i  §-x;     ^-y,  z+i  x+|; 

y+i  l-z,  x+^; 
y+i  x+i  z+l;     |-y,  x+|,  |-z;    y+f,  |-x,  |-z; 

l-y,  i-x,  z+§; 
x+i  z+i  y+J;    x+i  ^-z,  |-y;    |-x,  ^-z,  y-(-^; 

l-x,  z+i  l-y; 
z+i  y+i  x-l-^;    ^-z,  |-y,  x+^;    ^-z,  y+l,  A-x; 

z+i  ^-y,  5-x; 

x+i  y+i  z;  x+i  ^-y,  z;  |-x,  y-|-|,  z;  f-x,  |-y,  z; 

z+i  x+l,  y;  |-z.  x+i  y;  ^-z,  ^-x,  y;  z+|,  |-x,  y; 

y+i  z+i  x;  |-y,  |-z,  x;  y+i,  |-z,  x;  |-y,  z+^,  x; 

l-y,  ^-x,  z;  y+^,  |-x,  z;  |-y,  x+i,  z;  y+|,  x+^,  z; 

|-x,  l-z,  y;  i-x,  z+i  y;  x+i  ^+^,  y;  x+i  |-z,  y; 

^-z,  l-y,  x;  z+i  y+i  x;  z+|,  i-y,  x;  ^-z,  y+i  x; 

X,  y,  ^-z;  X,  y,  z+^;  x,  y,  z+^;  x,  y,  |-z; 

z,  X,  ^-y;  z,  X,  y+|;  z,  x,  |-y;  z,  x,  y+|; 

y,  z,  l-x;  y,  z,  |-x;  y,  z,  x+^;  y,  z,  x+|; 

y,  X,  z+§;  y,  x,  ^-z;  y,  x,  ^-z;  y,  x,  z+|; 

X,  z,  y+l;  x,  z,  |-y;  x,  z,  y+|;  x,  z,  |-y; 

z,  y,  x-f-l;  z,  y,  x-l-i;  z,  y,  |-x;  z,  y,  |-x; 

x+i  y,  z+l;  x+i  y,  ^-z;  ^-x,  y,  |-z;  ^-x,  y,  z+^; 

z+i  X,  y+l;  i-z,  X,  ^-y;  |-z,  x,  y+|;  z+|,  x,  ^-y; 

y+iz,  x+^;  ^-y,  z,  x+l;  y+|,  z,  ^-x;  |-y,  z,  |-x; 

h-y,  X,  h-z;  y+i  x,  z+l;  |-y,  x,  z+|;  y+i  x,  ^-z; 

^-x,  z,  ^-y;  l-x,  z,  y+|;  x+§,  z,  ^-y;  x+§,  z,  y+^; 

^-z,  y,  |-x;  z+i  y,  |-x;  z-f-^  y,  x-F§;  ^-z,  y,  x+|; 


144 


THE   CUBIC   SPACE-GROUPS  0^-0^. 


Space-Group  0^  (continued). 


X,  l-y, 

z; 

X,  y+l,  z; 

X,  I-y,  z; 

X,  y+l. 

z; 

z,  l-x, 

y; 

z,  §-x,  y; 

z,  x+l,  y; 

z,  x+l, 

y; 

y,  ^-z, 

x; 

y,  z+§,  x; 

y,  z+l,  x; 

y,  l-z, 

x; 

y,  x+l, 

z; 

y,  x+l,  z; 

y,  l-x,  z; 

y,  l-x. 

z; 

X,  z  +  l, 

y; 

X,  i-z,  y; 

X,  l-z,  y; 

X,  z+l, 

y; 

z,  y+i 

x; 

z,  I-y,  x; 

z,  y+l,  x; 

z,  I-y, 

x; 

X,  y+i 

z+l; 

X,  h-y,  l-z; 

X,  y+l,  l-z; 

X,  I-y, 

z+l 

Z,   x+2, 

y+l; 

z,  x+i  I-y; 

z,  |-x,  y+l; 

Z,    2       X, 

I-y 

y,  z+i 

x+l; 

y?    2        Z,   X+2i 

y,  l-z,  |-x; 

y,  z+l, 

l-x 

y,  ¥-x, 

l-z; 

y,  l-x,  z+l; 

y,  x+l,  z+l; 

y,  x+l, 

l-z 

X,  ^-z, 

i-y; 

X,  z+l,  y+l; 

X,  z+l,  I-y; 

X,  l-z, 

y+l 

z,  i-y, 

1  — V 

2  X, 

z,  y+l,  l-x; 

z,  I-y,  x+l; 

z,  y+l, 

x+l 

i-x,  y, 

z; 

l-x,  y,  z; 

x+l,  y,  z; 

x+l,  y, 

z; 

^-z,  X, 

y; 

z+l,  X,  y; 

z+l,  X,  y; 

l-z,  x. 

y; 

i-y,  z, 

x; 

y+l,  z,  x; 

I-y,  z,  x; 

y+l,  z. 

x; 

y+l,  X, 

z; 

i-y,  X,  z; 

y+l,  X,  z; 

I-y,  X, 

z; 

x+i  z, 

y; 

x+l,  z,  y; 

l-x,  z,  y; 

l-x,  z, 

y; 

z+i  y, 

x; 

l-z,  y,  x; 

l-z,  y,  x; 

z+l,  y. 

X. 

Space-Group  0^. 

Eight  equivalent  positions: 

(a)  8f.  (b)  8g. 

Sixteen  equivalent  positions : 

(c)  16b.  (d)  16c. 

Thirty-two  equivalent  positions : 

(e)  32b. 

Forty-eight  equivalent  positions : 

(f)  48c. 

Ninety-six  equivalent  positions : 

(g)  96e.  (h)  96f. 

One  hundred  ninety-two  equivalent  positions: 
(i)  xyz;        xyz;        xyz;        xyz; 
zxy;        zxy;        zxy;        zxy; 
yzx;        yzx;        yzx;        yzx; 


-y, 


i_. 


:-z; 


y+ 


1    i_' 


A  X,        A  Z, 


t-z,  t-y. 


y; 


—  x. 


X,  z+i;    i-y,  x+i  z+i; 

y+4,  x+j,  4  — z; 
l-x,  z+i  y+i;    x+i  z+i,  i-y; 

x+i  l-z,  y+i; 
i-x;    z+i,  y+i,  i-x;    z+i,  i-y,  x+i; 

i-z,  y+i,  x+i; 
i-z;    i-x,  y+i,  z+i;    x+i,  i-y,  z+i; 

x+i,  y+i,  i-z; 


4 


THE   CUBIC   SPACE-GROUP   0^. 


145 


Space-Group  O^  {continued). 

1—7     i  — V     i  — 

4        ^)     4        ^)     i 


4        Jj     4        ^>     4 

yxz;        yxz; 
xzy;        xzy; 
zyx;        zyx; 
x+i  y+i  z; 
z+i  x+i  y; 
y+i  z+l,  x; 
4~y>   4~X,  j- 


y;   z+i,  i-x, 
x;   y+i  z+i 


y+l;     z+i  x+l,  |-y; 

l-z,  x+i  y+i; 
4~x;     4~y,  z+4,  x+j; 

y+i  ?-z,  x+l; 


yxz;  yxz; 
xzy;  xzy; 
zyx;         zyx; 

x+i  |-y,  z;     l-x,  y+^z;     i-x,  |-y,  z; 
^-z,  x+i  y; 
i-y,  h-z,  x; 
■z;    y+i  f-x, 


3_v      1 
4        *>     4 


•z,  t-y 


1—7      1  — V       i 

4        ^>     4        J>     4 


i-x 


4        X,     4        y>     4 


—  z 


3_7 
4        Z, 


■X,     4 


-y 


4      y>    4      z,    4      X 


l-z,  I-x,  y;     z+l,  I-x,  y; 
y+i  l-z,  x;     |-y,  z+l,  x; 

z+4j   4— y>  x+4,  z+4; 


y+i  x+l,  z; 
x+l,  z+l,  y; 
z+l,  y+l,  x; 
x+l,  y,  z+l; 
z+l,  X,  y+l; 
y+l,  z,  x+l; 

f-y 


y+i  x+f,  t-z; 
y+i;     x+f,  z+f,  i-y; 

x+f,  f-z,  y+i; 
4  — x;     z+4,  4— y,  x+j; 

4~z,  y+4,  x+j; 
f-x,  y+f,  z+i;     x+f,  f-y,  z+i; 

x+4,  y+4,  4  — z; 
y+i;    z+f,  x+f,  i-y; 

4  — z,  x+4,  y+y; 
4— x;    4— y,  z+4,  x+4; 

y+i  f-z,  x+i; 


4        X,    z+4, 

z+i  y+i 


Z+4J     4— X, 

y+i  z+i 


1_Y       1. 

4  A,      4 


l-y,  x+l,  z 

x+l,  l-z,  y 
^-z,  |-y,  x: 
x+l,  y,  l-z 
2  — z,  X,  ^— y 
|-y,  z,  x+l 
z;    y+i  i-x. 


4— X,  4  — z,  4     y 


■z,  t-y. 


1_Y      i  — 
4        A,     4 


y,  f 


—  z 


1—7       1_Y 
4        ^>     4        X, 


y>     4         Z,     4 


4        X,     Z+4, 

z+i  y+i 
f-x,  y+i, 

z  +  4,    4— X, 


f-x;    y+i  z+i 


y+l,  X,  z+l 
x+l,  z,  y+l 
z+l,  y,  x+l 
X,  y+l,  z+l 

z,  x+l,  y+l 


l-y,  X,  l-z; 
x+l,  z,  |-y; 
2— z,  y,  x+2; 
X,  l-y,  l-z; 
z,  x+l,  l-y; 


y+i  I-x,  z;  l-y,  I-x,  z; 
I-x,  l-z,  y;  I-x,  z+l,  y; 
l-z,  y+l,  x;  z+l,  l-y,  x; 
I-x,  y,  l-z;  I-x,  y,  z+|; 
l-z,  X,  y+l;  z+l,  X,  l-y; 
y+i  z,  |-x;  l-y,  z,  I-x; 
z+f;    f-y,  x+i  z+f; 

y+4j  x+4,  4  — z; 
y+f;     x+i  z+i  f-y; 

x+i  i-z,  y+f; 
f-x;     z+i  i-y,  x+f; 

4  — z,  y+4,  x+j; 
z+f;    x+i  f-y,  z+f; 

x+i  y+i  f-z; 
y+f;    z+i  x+i  f-y; 

f-z,  x+i  y+f; 
f-x;    f-y,  z+i  x+f; 

y+i  i-z,  x+f; 
y+i  X,  l-z;  l-y,  x,  z+|; 
I-x,  z,  y+l;  I-x,  z,  |-y; 
2  — z,  y,  ^— x;  z+2,  y,  2— x; 
X,  y+i  l-z;  X,  l-y,  z+|; 
z,  2-x,  y+l;    z,  |-x,  l-y; 


146  THE   CUBIC  SPACE-GROUPS  0^-0^. 

Space-Group  Ol  {continued). 

y,  z+i  x+l;  y,  ^-z,  x+|;  y,  |-z,  ^-x;  y,  z+|,  ^-x; 
4~y>  4~x,  4  — z;    y+i,  4— X,  z+4;    4  — y,  x-f-j,  z+4; 

y+jj  x~|-4,  4~z; 
i"~x,  4— z,  4— y;    4~x,  Z-I-4,  y4-4;    x+j,  z+4,  4— y; 

x+4)  4  2,  y+4i 
l-z,  f-y,  f-x;     z+i,  y+f,  |-x;     z+|,  f-y,  x+f; 

l-z,  y+i  x+l; 
l-x,  l-y,  l-z;     i-x,  y-f-f,  z+i;    x+i  f-y,  z+f; 

^+4>  y+4j  4~z; 
j-z,  f-x,  f-y;     z+i  f-x,  y+f;     z+|,  x+f,  f-y; 

4  — z,  x+4,  y+i; 
f-y,  f-z,  f-x;     y+f,  z+f,  f-x;     f-y,  z+f,  x+f; 

y+4>  4""^,  x+4; 

y,  x+l,  z+l;  y,  x+|,  ^-z;  y,  |-x,  |-z;  y,  i-x,  z+|; 
X,  z+2,  y+2;  X,  2~z,  ^— y;  x,  2~z,  y+2l  X,  z+^,  2~y5 
z,  y+i  x+l;    z,  |-y,  x+^;    z,  y+|,  l-x;    z,  |-y,  |-x. 

Space-Group  O*. 

Sixteen  equivalent  positions: 

(a)  16h. 

Thirty-two  equivalent  positions : 

(b)  32d.  (c)  32e. 
Forty-eight  equivalent  positions : 

(d)  48i. 

Sixty-four  equivalent  positions: 

(e)  64b. 

Ninety-six  equivalent  positions: 

(f)  96g.  (g)  96h. 

One  hundred  ninety-two  equivalent  positions: 
(h)  xyz;        xyz;        xyz;        xyz; 
zxy;        zxy;        zxy;        zxy; 
yzx;        yzx;        yzx;        yzx; 
f-y,  f-x,  f-z;    y+f,  f-x,  z+f;     f-y,  x+f,  z+f; 

y+4,  x+4,  4  — z; 
f-x,  f-z,  f-y;     f-x,  z+f,  y+f;     x+f,  z+f,  f-y; 

x+f,  f-z,  y+f; 
f-z,  f-y,  f-x;    z+f,  y+f,  f-x;     z+f,  f-y,  x+f; 

f-z,  y+f,  x+f; 
f-x,  f-y,  f-z;    f-x,  y+f,  z+f;    x+f,  f-y,  z+f; 

x+f,  y+f,  f-z; 
f-z,  f-x,  f-y;    z+f,  f-x,  y+f;    z+f,  x+f,  f-y; 

f-z,  x+f,  y+f; 


THE   CUBIC   SPACE-GROUP   OJ.  147 

Space-Group  0'  {contirvaed). 

l-y,  l-z,  l-x;    y+l,  z+f,  |-x;     |-y,  z+f,  x+f; 

y+i  l-z,  x+l; 
y+i  x+i  z+^;    |-y,  x+i  \-z;    y+i  ^-x,  |-z; 

l-y,  l-x,  z+l; 
x+i  z+l,  y+l;    x+l,  l-z,  |-y;    |-x,  |-z,  y+|; 

2~~x,  z+2,  2~yj 

z+i  y+l,  x+l;     l-z,  |-y,  x+|;     |-z,  y+|,  |-x; 

z+i,  ^  — y>  i"  — x; 
x+l,  y+l,  z;     x+l,  ^-y,  z;     |-x,  y+|,  z;    |-x,  |-y,  z; 
z+l,  x+l,  y;     l-z,  x+iy;     |-z,  |-x,  y;     z+|,  ^-x,  y; 
y+l,  z+l,  x;     l-y,  l-z,  x;     y+|,  |-z,  x;     ^-y,  z+i  x; 
f-y,  l-x,  i-z;     y+f,  l-x,  z+i;     f-y,  x+f,  z+i; 

y+4,  x+4,  J  — z; 
l-x,  l-z,  i-y;     f-x,  z+l,  y+i;     x+f,  z+|,  i-y; 

x+4,  4  — z,  y+4j 
4  — z,  4— y,  4  — x;    z+4,  y+4,  4— x;    z+j,  4— y,  x+j; 

4— z,  y+4,  x+jj 
i-x,  i-y,  i-z;    i-x,  y+i  z+|;    x+i,  f-y,  z+|; 

x+i,  y+i  l-z; 
i-z,  i-x,  l-y;     z+i  i-x,  y+|;     z+i  x+i  f-y; 

4  — z,  x+j,  y+4; 
i-y,  i-z,  l-x;    y+i  z+i  f-x;    i-y,  z+i  x+|; 

y+i  i-z,  x+l; 
y,  X,  z+l;         y,  x,  |-z;         y,  x,  |-z;        y,  x,  z+|; 
X,  z,  y+l;  X,  z,  |-y;  x,  z,  y+|;         x,  z,  |-y; 

z,  y,  x+l;         z,  y,  x+l;         z,  y,  ^-x;         z,  y,  |-x; 
x+l,  y,  z+l;    x+l,  y,  i-z;     |-x,  y,  |-z;    |-x,  y,  z+|; 
z+l,  X,  y+l;     l-z,  X,  |-y;    f-z,  x,  y+|;     z+|,  x,  ^-y; 
y+i  z,  x+l;     l-y,  z,  x+|;     y+i  z,  |-x;     |-y,  z,  |-x; 
l-y,  i-x,  l-z;     y+f,  i-x,  z+|;     f-y,  x+i  z+f; 

y+i  x+i  f-z; 
f-x,  i-z,  f-y;     f-x,  z+i  y+f;     x+i  z+i  f-y; 

x+i  i-z,  y+f; 
f-z,  i-y,  f-x;     z+i  y+i  f-x;     z+i  i-y,  x+f; 

f-z,  y+i  x+f; 
i-x,  f-y,  i-z;     i-x,  y+i  z+f;     x+i,  f-y,  z+f; 

x+i  y+i  i-z; 
i-z,  f-x,  i-y;     z+i  f-x,  y+f;     z+i  x+i  i-y; 

\-z,  x+i  y+i; 
i-y,  l-z,  i-x;    y+i  z+i  i-x;    i-y,  z+i  x+i; 

y+i  f-z,  x+i; 
y,  x+l,  z;         y,  x+l,  z;         y,  ^-x,  z;        y,  ^-x,  z; 
X,  z+i  y;         x,  l-z,  y;         x,  |-z,  y;        x,  z+|,  y; 
z,  y+l,  x;  z,  l-y,  x;  z,  y+i  x;         z,  |-y,  x; 

X,  y+l,  z+i;     X,  l-y,  l-z;     x,  y+|,  |-z;    x,  |-y,  z+|; 


148  THE    CUBIC   SPACE-GROUPS   0^-02- 

Space-Group  0^  (continued). 

z,  x+iy+l;  z,  x+i^-y;  z,  |-x,  y+l;  z,  |-x,  |-y; 
y,  z+i  x+l;  y,  f-z,  x+^;  y,  |-z,  ^-x;  y,  z+i  ^-x; 
i-y,  f-x,  f-z;    y+i  f-x,  z+f;     i-y,  x+f,  z+f; 

y+h  x+i  l-z; 
l-x,  f-z,  l-y;     l-x,  z+l,  y+f;     x+i,  z+f,  f-y; 

x+i  f-z,  y+f; 
l-z,  f-y,  f-x;     z+l,  y+f,  f-x;     z+|,  f-y,  x+f; 

i-z,  y+i  x+f; 
f-x,  i-y,  i-z;     f-x,  y+i,  z+|;    x+f,  |-y,  z+f; 

x+i  y+i  f-z; 
f-z,  f-x,  f-y;     z+f,  f-x,  y+f;     z+f,  x+f,  f-y; 

■J— z,  x+4,  y+j; 
l-y,  i-z,  f-x;     y+f,  z+f,  f-x;     f-y,  z+f,  x+f; 

yij,  4  — z,  x+j; 
y+l,  X,  z;     l-y,  x,  z;     y+|,  x,  z;     ^-y,  x,  z; 
x+i  z,  y;     x+l,  z,  y;     ^-x,  z,  y;     |-x,  z,  y; 
z+i  y,  x;     i-z,  y,  x;     f-z,  y,  x;     z+f,  y,  x. 

Space-Group  O^. 

Two  equivalent  positions: 

(a)  2a. 

Six  equivalent  positions : 

(b)  6e. 

Eight  equivalent  positions: 

(c)  8e. 

Twelve  equivalent  positions: 

(d)  12h.  (e)  12a. 
Sixteen  equivalent  positions: 

(f)  16d. 

Twenty-four  equivalent  positions : 

(g)  24f.  (h)  24j. 
Forty-eight  equivalent  positions: 

(i)  481.  (j)  48j.  (k)  48k. 

Ninety-six  equivalent  positions: 

(1)     xyz;  xyz;  xyz;  xyz; 

zxy;  zxy;  zxy;  zxy; 

yzx;  yzx;  yzx;  yzx; 

yxz;  yxz;  yxz;  yxz; 

xzy;  xzy;  xzy;  xzy;  • 


THE   CUBIC   SPACE-GROUPS  0^- 


Ufa. 


149 


Space-Group  Oh  (continued). 


zyx; 

zyx; 

zyx; 

zyx; 

xyz; 

xyz; 

xyz; 

xyz; 

zxy; 

zxy; 

zxy; 

zxy; 

yzx; 

yzx; 

yzx; 

yzx; 

yxz; 

yxz; 

yxz; 

yxz; 

xzy; 

xzy; 

xzy; 

xzy; 

zyx; 

zyx; 

zyx; 

zyx; 

x+l,  y+i  z+§ 
z+i,  x+i  y+l 
y+l,  z+i  x+l 
2~y>  2~x,  ^— z 


|-x,  §-z,  l-y 


2  — z,  5— y,  2~x 
l-x,  |-y,  i-z 
l-z,  i-x,  §-y 


l-y,  ^-z,  ^-x 


y-hh  x+l,  z+i 
x+i  z+i  y+l 
z+i  y+i  x+l 

Space-Group  0^*. 

Sixteen  equivalent  positions: 
(a)  16h.  (b)  16i. 

Twenty-four  equivalent  positions : 
(c)  24v.  (d)  24w. 

Thirty-two  equivalent  positions : 

(e)  32f. 

Forty-eight  equivalent  positions : 

(f)  48m.  (g)  48n. 


X+2J  2~y>  2~Z 
2~z,  x+2,  2~y 
h-y,  §-z,  x-M 
y+2>  2~x,  z4-2 

^-x,  z+l,  y+l 
z+i  y+h  ¥-x 
l-x,  y+i  z+l 
z+i  ^-x,  y-{-^ 

y+2>    Z+2,     2~X 

2~y>  X4-2j  ?~z 
x+i  l-z,  l-y 

2~Z,     2~y)    X-f-2 


2~x,  y+2)  2~z; 
2"~x,  2~y)  z-j-2; 

2~Z,    2'~X,    y+2; 

Z+2>    2~X,    2~yj 

y  I  2)  2~z,   2~x; 

2~y>  z-f-2>  2~x; 
2~y>  x+2>  z+j-; 

y+i  x+l,  l-z; 
^+h  z+i  |-y; 

x+2)     2~Z,    y+2; 

z+2>  ^~y)  x+2"; 

2~z,  y+i,  x+2; 
x+i,  2~y)  z+2; 

x+i  y+i  |-z; 

Z+2>    X+2,     2~y; 

^-z,  3C+5,  y+^; 
2~y>  z+2,  x+2; 
yr2,  2~z,  x+2; 

y~r2>  2^~x,  2~z; 

h-y,  l-x,  z+^; 
^-x,  i-z,  y+l; 

i-x,  z+i  ^-y; 
^-z.  y+i  i-x; 

z+i  ^-y,  i-x. 


150  THE   CUBIC  SPACE-GROUP  0^. 

Space-Group  0^°  (continued). 

Ninety-six  equivalent  positions: 

(h)  xyz;  x,  y,  ^-z;  |-x,  y,  z;  x,  |-y,  z; 
zxy;  i-z,  X,  y;  z,  ^-x,  y;  z,  x,  ^-y; 
yzx;  y,  |-z,  x;  y,  z,  |-x;  ^-y,  z,  x; 
j-y>  l-x,  i-z;    y+i,  i-x,  z+f;    J-y,  x+f,  z+i; 

y"r4>  x+4,  j  —  z', 
l-x,  l-z,  i-y;    i-x,  z+f,  y+i;    x+f,  z+i  f-y; 

x+i,  i-z,  y+f ; 
i-z,  i-y,  i-x;     z+f,  y+f,  f-x;    z+f,  f-y,  x+f; 

f-z,  y+f,  x+f; 
xyz;    X,  y,  z+|;    x+i  y,  z;    x,  y+|,  z; 
zxy;     z+i  X,  y;     z,  x+i  y;     z,  x,  y+|; 
yzx;    y,  z+|,  x;     y,  z,  x+|;     y+|,  z,  x; 
y+f,  x+f,  z+f;     f-y,  x+f,  f-z;    y+f,  f-x,  f-z; 

f-y,  f-x,  z+f; 
x+f,  z+f,  y+f;    x+f,  f-z,  f-y;    f-x,  f-z,  y+f; 

4~x,  z+j,  4— y; 
z+f,  y+f,  x+f;     f-z,  f-y,  x+f;     f-z,  y+f,  f-x; 

z+4,  4— y,  4— x; 
x+l,  y+i  z+l;    x+i  |-y,  z;  x,  y+i  |-z; 

i-x,  y,  z+l; 
z+i  x+i  y+i;    z,  x+i  |-y;  |-z,  x,  y+|; 

z+i,  l-x,  y; 
y+i  z+§,  x+l;    |-y,  z,  x+l;  y+i  |-z,  x; 

y,  z+2,  2— x; 
f-y,  f-x,  f-z;    y+f,  f-x,  z+f;    f-y,  x+f,  z+f; 

y+f,  x+f,  f-z; 
f-x,  f-z,  f-y;    f-x,  z+f,  y+f;    x+f,  z+f,  f-y; 

x+f,  f-z,  y+f; 
f-z,  f-y,  f-x;    z+f,  y+f,  f-x;    z+f,  f-y,  x+f; 

f-z,  y+f,  x+f; 
l-x,  |-y,  |-z;    l-x,  y+l,  z;  x,  |-y,  z+|; 

x+l,  y,  |-z; 
l-z,  l-x,  l-y;     z,  l-x,  y+|;  z+|,  x,  |-y; 

|-z,  x+l,  y; 
l-y,  l-z,  l-x;    y+l,  z,  l-x;  |-y,  z+|,  x; 

y,  |-z,  x+l; 
y+f,  x+f,  z+f;    f-y,  x+f,  f-z;    y+f,  f-x,  f-z; 

f-y,  f-x,  z+f; 
x+f,  z+f,  y+f;    x+f,  f-z,  f-y;    f-x,  f-z,  y+f; 

f-x,  z+f,  f-y; 
z+f,  y+f,  x+f;    f-z,  f-y,  x+f;    f-z,  y+f,  f-x; 

z+f,  f-y,  f-x. 


THE   HEXAGONAL   SPACE-GROUPS   C3-C3I.  151 

HEXAGONAL  SYSTEM. 
RHOMBOHEDRAL  DIVISION. 
A.  TETARTOHEDRY. 
Space-Group  CJ. — (Hexagonal  Axes.) 
One  equivalent  position: 

(a)OOu.  (b)Hu.  (c)Hii. 

Three  equivalent  positions: 

(d)  xyz;    y-x,  x,  z;    y,  x-y,  z. 
Space-Group  C3. — (Hexagonal  Axes.) 

Three  equivalent  positions: 

(a)  xyz;    y-x,  x,  z-f-f;    y,  x-y,  z+f, 
Space-Group  C|. — (Hexagonal  Axes.) 
Three  equivalent  positions: 

(a)  xyz;    y-x,  x,  z+f;    y,  x-y,  z+f 
Space-Group  C3. — (Rhombohedral  Axes.) 
One  equivalent  position: 

(a)  uuu. 

Three  equivalent  positions: 

(b)  xyz;    zxy;    yzx. 

B.  HEXAGONAL  TETARTOHEDRY  OF  THE  SECOND  SORT. 
Space-Group  C3,. — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  000.  (b)  00|. 

Two  equivalent  positions: 

(c)  OOu;    GOii.  (d)  Hu;    f  i  u- 
Three  equivalent  positions: 

(e)  iH;    OH;    iOl  (f)  HO;    O^O;    HO. 

Six  equivalent  positions: 

(g)  xyz;    y-x,  x,  z;    y,  x-y,  z; 
xyz;    x-y,  x,  z;    y,  y-x,  z. 

Space-Group  C|,. — (Rhombohedral  Axes.) 

One  equivalent  position: 

(a)  000.  (b)Hi 


152  THE   HEXAGONAL   SPACE-GROUPS   C3I-C3V. 

Space-Group  Ca^i  (continued). 
Two  equivalent  positions: 

(c)  uuu;    tiiiii. 
Three  equivalent  positions: 

(d)OO^;    ^00;    0^0.  (e)  H  0;    OH;    iO^ 

Six  equivalent  positions : 

(f)   xyz;    zxy;    yzx;        xyz;    zxy;    yzx. 

C.  HEMIMORPHIC  HEMIHEDRY. 
Space-Group  Cay. — (Hexagonal  Axes.) 
One  equivalent  position: 

(a)OOu.  (b)  Hu.  (c)f|u. 

Three  equivalent  positions: 

(d)  utiv;    2u,  u,  v;    u,  2u,  v. 
Six  equivalent  positions : 

(e)  xyz;    y-x,  x,  z;    y,  x-y,  z; 
yxz;    X,  x-y,  z;    y-x   y    z. 

Space-Group  Cay. — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  OOu. 

Two  equivalent  positions : 

(b)  Hu;    Hu. 
Three  equivalent  positions: 

(c)  uuv;    Ouv;    ti  0  v. 
Six  equivalent  positions: 

(d)  xyz;    y-x,  x,  z;    y,  x-y,  z; 
yxz;    x,  y-x,  z;    x-y,  y,  z. 

Space-Group  CaV — (Hexagonal  Axes.) 

Two  equivalent  positions: 

(a)  OOu;    0,  0,  u-Fi  (c)  H  u;    f ,  I  u-f-f 

(b)  Hu;    i  f,  n-\-h 

Six  equivalent  positions: 

(d)  xyz;  y-x,  x,  z;    y,  x-y,  z; 

X,  x-y,  z+^;    y,  x,  z+i;    y-x,  y,  z-H. 


THE   HEXAGONAL   SPACE-GROUPS  Csy-Ds-  153 

Space-Group  C*^. — (Hexagonal  Axes.) 

Two  equivalent  positions : 

(a)OOu;    0,  0,  u+i  (b)Hu;     f,  i  u+i. 

Six  equivalent  positions : 

(c)  xyz;  y-x,  x,  z;  y,  x-y,  z; 

y,  X,  z+i;    X,  y-x,  z+l;    x-y,  y,  z+|. 

Space-Group  Cat- — (Rhombohedral  Axes.) 

One  equivalent  position: 

(a)  u  u  u. 

Three  equivalent  positions : 

(b)  uu  v;    vuu;    u  vu. 
Six  equivalent  positions: 

(c)  xyz;    zxy;    yzx;        xzy;    Z3rx;    yxz. 
Space-Group  Cay. — (Rhombohedral  Axes.) 

Two  equivalent  positions: 

(a)  uuu;    u+l,  u-M,  u-|-^. 
Six  equivalent  positions: 

(b)  xyz;  zxy;  yzx; 

x+i  z+i  y-|-|;    z-M,  y-hi  x+^;    y-f-i  x-Hi  z+|. 

D.  ENANTIOMORPHIC  HEMIHEDRY. 
Space-Group  D3. — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  000.  (c)  HO.  (e)  f  |  0. 

(b)OOi  (d)Hi  (f)  IH. 

Two  equivalent  positions: 

(g)  OOu;    OOu.  (i)   Hu;    H  u. 

(h)Hu;    HQ. 

TAree  equivalent  positions : 

(j)  uuO;    2u,  u,  0;    u,  2u,  0. 
(k)  ua^;    2u,  u,  I;    u,  2u,  ^. 

<Sfa;  equivalent  positions: 

(1)   xyz;  y-x,  x,  z;    y,  x-y,  z; 

x,  x-y,  z;    yxz;  y-x,  y,  z. 


154  THE   HEXAGONAL   SPACE-GROUPS  D3-D3. 

Space-Group  D3. — (Hexagonal  Axes.) 
One  equivalent  position: 

(a)  000.  (b)  OOi 

Two  equivalent  positions: 

(c)  OOu;    OOu.  (d)  Hu;    IH- 

Three  equivalent  positions: 

(e)  uuO;    OuO;    tiOO.  (f)  uu|;    Ou|;    ti  0  §. 

Six  equivalent  positions: 

(g)  xyz;    y-x,  x,  z;    y,  x-y,  z; 
yxz;    X,  y-x,  z;    x-y,  y,  z. 

Space-Group  D|. — (Hexagonal  Axes.) 

Three  equivalent  positions: 

(a)  utii;    2u,  u,  f;    u,  2u,  0. 

(b)  uuf;    2u,  u,  I;    u,  2u,  ^. 

Six  equivalent  positions: 

(c)  xyz;  y-x,  x,  z-f^;    y,  x-y,  z-ff; 
y-x,  y,  z;    y,  x,  f-z;         x,  x-y,  f-z. 

Space-Group  D3. — (Hexagonal  Axes.) 

Three  equivalent  positions: 

(a)  uOO;    uui;    Ouf.  (b)  uO^;    Quf;    Ouf. 

Six  equivalent  positions: 

(c)  xyz;  y-x,  x,  z+^;    y,  x-y,  z+f; 

x-y,  y,  z;    y,  X,  |-z;  x,  y-x,  ^-z. 

Space-Group  D3. — (Hexagonal  Axes.) 

Three  equivalent  positions: 

(a)  uu|;    2u,  u,  |;    u,  2u,  i. 

(b)  utif;    2u,  u,  ^;    u,  2u,  0. 

Six  equivalent  positions: 

(c)  xyz;  y-x,  x,  z+f;    y,  x-y,  z-}-^; 
y-x,  y,  z;    y,  x,  f-z;  x,  x-y,  f-z. 

Space-Group  D*. — (Hexagonal  Axes.) 
Three  equivalent  positions: 

(a)  uOO;    Out;    utif.  (b)uO^;    Ouf;    Qui 

Six  equivalent  positions: 

(c)  xyz;  y-x,  x,  z+f;    y,  x-y,  z+l; 

x-y,  y,  z;    y,  x,  ^-z;         x,  y-x,  f-z. 


THE   HEXAGONAL   SPACE-GROUPS  Dj-dIj.  155 

Space-Gboup  D3. — (Rhombohedral  Axes.) 
One  equivalent  position : 

(a)  0  0  0.  (b)  H  h 

Two  equivalent  positions: 

(c)  u  u  u;    ti  u  u. 
Three  equivalent  positions : 

(d)  uuO;    tiOu;    Ouu.  (e)  uu^;    u|u;    ^uti. 
Six  equivalent  positions: 

(f)   xyz;    yzx;    zxy;    yxz;    xzy;    zyx. 

E.  HOLOHEDRY. 
Space-Group  Dl^. — (Hexagonal  Axes.) 
One  equivalent  position: 

(a)  0  0  0.  (b)  OOi 

Two  equivalent  positions: 

(c)  HO;    fiO.  (e)  OOu;    0  0  u. 

/JN     13    1.211 
\^)     3  5  2  J        3    3   2- 

Three  equivalent  positions: 

(f)  HO;   o|0;   ^00.  (g)  Hi;   OH;   ^0^. 

Four  equivalent  positions: 

/U^     12,1.        1    2  ,-,  .        21,,.        2    1  fi 
Wl^U,        ^gU,        33U,        3lU. 

Six  equivalent  positions: 

(i)  uiiO;  2u,  ti,  0;  u,  2u,  0;  uuO;  2u,  u,  0;  u,  2u,  0. 
(j)  uui;  2u,  ti,  ^;  u,  2u,  i;  uu|;  2u,  u,  ^;  ti,  2u,  ^ 
(k)  uuv;    Otiv;    u  0  v;    uOv;    uuv;    Ouv. 

Twelve  equivalent  positions: 

(1)    xyz;    y-x,  x,  z;    y,  x-y,  z;    x,  x-y,  z;    y.xz;    y-x,  y,  z; 
xyz;    x-y,  x,  z;    y,  y-x,  z;    x,  y-x,  z;    yxz;    x-y,  y,  z. 

Space-Group  Dgj. — (Hexagonal  Axes.) 

Two  equivalent  positions : 

(a)  000;    OOi  (c)  HO;    f  H- 

(b)OOi;    OOf.  (d)Hi;    fio. 

Four  equivalent  positions : 

(e)  OOu;    OOti;    0,  0,  ^-u;    0,  0,  u+i 

(f)  Hu;    Hti;    f,  h  l-u;    f,  i  u-M. 


y-x, 

X,  z; 

y,  X- 

-y, 

z; 

yxz; 

y-x, 

y, 

z; 

x-y, 

X,  l-z; 

y,  y- 

-X, 

i-z 

y,  X, 

z+^; 

x-y, 

y, 

z+l 

156  THE  HEXAGONAL  SPACE-GROUPS  D^-Dm- 

Space-Group  D^  (continued). 

Six  equivalent  positions : 

(a\  111'    nil-    ini-     in3'     113.    nis 
(h)  uuO;    2u,  u,  0;    u,  2u,  0;    uu|;    2u,  u,  ^;    Q,  2u,  §. 
Twelve  equivalent  positions: 
(i)   xyz; 

X,  x-y,  z; 
X,  y,  i-z; 
X,  y-x,  z+l; 

Space-Group  D'a. — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  0  0  0.  (b)  0  0  i 

Two  equivalent  positions: 

(c)  OOu;    OOti.  (d)  Hu;    Hu. 

Three  equivalent  positions : 

(e)  HO;    OiO;    ^00.  (f)  Hi;    OH;    ioi 

Six  equivalent  positions : 

(g)  uuO 
(h)  uu| 
(i)   u  ti  V 

Twelve  equivalent  positions: 

(j)   xyz;                y-x,  x,  z;    y,  x-y,  z; 

X,  y-x,  z;    yxz;  x-y,  y,  z; 

xyz;               x-y,  x,  z;    y,  y-x,  z; 

X,  x-y,  z;    yxz;  y-x,  y,  z. 

Space-Group  T>za- — (Hexagonal  Axes.) 
Two  equivalent  positions: 

(a)  0  0  0;    0  0|.  (b)  0  0^;    0  0  f . 

Four  equivalent  positions: 

(c)  OOu;    OOQ;    0,  0,  ^-u;    0,  0,  u-hi 

(d)Hu;    fiti;    i  i  ^-u;    i  I,  u-f-i 
Six  equivalent  positions: 

(e)  OH;   |0i;   Hi;    oH;   iof;   Hi 

(f)  uuO;    OuO;    uOO;    uu^;    Ou|;    uOi 
Twelve  equivalent  positions : 

(g)  xyz;  y-x,  x,  z;  y,  x-y,  z; 
X,  y-x,  z;  yxz;  x-y,  y,  z; 
X,  y,  ^-z;  x-y,  x,  ^-z;  y,  y-x,  ^-z; 
X,  x-y,  z+i;  y,  x,  z-F^;  y-x,  y,  z-f^. 


OuO;  tiOO;  uuO;  OuO;  u  0  0. 
Ou|;  uO^;  uu^;  Ou^;  u  0 1. 
2u,  u,  v;    u,  2u,  v;    uuv;    2u,  u,  v;    u,  2u,  v. 


THE   HEXAGONAL   SPACE-GROUPS  0^-0^.  157 

Space-Group  Di^. — (Rhombohedral  Axes.) 
One  equivalent  position: 

(a)  0  0  0.  (b)  H  i 

Two  equivalent  positions: 

(c)  uuu;    uuti. 
Three  equivalent  positions: 

(d)OO^;    0|0;    ^00.  (e)  H  0;    |0i;    OH- 

Six  equivalent  positions: 

(f)  utiO;    uOu;    Outi;    uuO;    uOu;    Otiu. 

(g)  uu|;    u|u;    ^uu;    tiu^;    u|u;    ^uu. 
(h)  uuv;    uvu;    vuu;    tiuv;    uvti;    vuu. 

Twelve  equivalent  positions: 

(i)    xyz;    yzx;    zxy;    yxz;    xzy;    zyx; 
xyz;    yzx;     zxy;     yxz;     xzy;     zyx. 

Space-Group  DgV — (Rhombohedral  Axes.) 
Two  equivalent  positions: 

(a)  000;    Hi  (b)iii;    f  f  f . 

Four  equivalent  positions : 

(c)  uuu;    uuu;    |-u,  |-u,  |-u;    u+|,  u+i,  u+^. 
Six  equivalent  positions : 

fr\\    133.        331.       113.       13,1.        3.11.        113 
\MJ    44  7>        444)        444)        444;        444)        44  f* 

(e)  uuO;  uOu;  Outi; 

l-u,  u-f-i  h;    u+l,  I,  |-u;    i,  l-u,  u-|-|. 

Twelve  equivalent  positions: 

(f)  xyz;        yzx;        zxy; 
yxz;        xzy;        zyx; 

i-x,  h-y,  |-z;    h-y,  l-z,  5-x;    §-z,  |-x,  ^-y; 
y+i  x-l-i  z+l;    x-hi  z-f-i  y-f-^;     z-|-^,  y-H,  x-H. 

HEXAGONAL  DIVISION. 
A.  TRIGONAL  PARAMORPHIC  HEMIHEDRY. 


Space-Group  Cgh. — (Hexagonal  Axes.) 

One  equivalent  position: 

(a)  000.                (c)  HO. 

(e)  UO. 

(b)00|.                (d)Hi 

(f)  IH. 

Two  equivalent  positions: 

(g)  OOu;    OOu.                (i)   \ 

liu;    Hu. 

(h)Hu;    HQ. 

158  THE  HEXAGONAL  SPACE-GROUPS   Cgh-D^. 

Space-Group  C^^  {continued). 
Three  equivalent  positions: 

(j)   uvO;    V— u,  u,  0;    v,  u— v,  0. 
(k)  uv^;    v-u,  u,  I;    v,  u-v,  |. 

Six  equivalent  positions : 

(1)    xyz;    y-x,  x,  z;    y,  x-y,  z; 
xyz;    y-x,  x,  z;    y,  x-y,  z. 

B.  HEMIHEDRY  WITH  A  THREE-FOLD  AXIS. 

{Trigonal  Holohedry.) 
Space-Group  DaV. — (Hexagonal  Axes.) 
One  equivalent  position: 


(a)  0  0  0.                (c)  H  0. 

(b)ooi.            (d)Hi 

(e)  f  1  0. 

Two  equivalent  positions: 

(g)  OOu;    OOu.                (i) 
(h)  Hu;    Hii. 

fiu;    fiu. 

Three  equivalent  positions: 

(j)   uuO;    2u,  u,  0;    u,  2u,  0. 

(k)  uu|;    2u,  u,  ^;    u,  2u,  ^. 

Six  equivalent  positions: 

(1)   uvO;  v— u,  u,  0;  v,  u— v,  0; 

u,  u— V,  0;    vuO;  v— u,  v,  0. 

(m)uv^;  v-u,  u,  i;  v,  u-v,  |; 

u,  u-v,  ^;    vu^;  v-u,  v,  ^. 

(n)  uuv;    2u,  u,  v;    u,  2u,  v;    uuv;    2u,  u,  v;    u,  2u,  v. 

Twelve  equivalent  positions: 

(o)  xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  x-y,  z;    yxz;  y-x,  y,  z; 

xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  x-y,  z;    yxz;  y-x,  y,  z. 

Space-Group  Da^u- — (Hexagonal  Axes.) 
Two  equivalent  positions : 


(a)  0  0  0; 

ooi 

(d)iH; 

Hi 

(b)OOi; 

OOf. 

(e)  fiO; 

IH. 

(c)  HO; 

Hi 

(f)  fH; 

fH. 

/^owr  equivalent  positions: 

(g)  OOu;    OOu;    0,  0,  ^-u;  0,  0,  u+i 

(h)Hu;    Hu;    i  f,  l-u;  i  I,  u-fi 

(i)    Hu;    IH;    f,  i  ^-u;  f,  i  u-fi 


THE   HEXAGONAL   SPACE-GROUPS  D^-Dgt-  159 

Space-Group  D^  (continued). 
Six  equivalent  positions : 

(j)   uuO;    2u,  u,  0;    u,  2u,  0;    uu|;    2u,  u,  ^;    u,  2u,  ^. 
(k)  uvi;  v-u,  u,  i;    v,  u-v,  i; 

u,  u-v,  I;    vuf;  v-u,  v,  f. 

Twelve  equivalent  positions: 

(1)    xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  x-y,  z;  yxz;  y-x,  y,  z; 

X,  y,  h-z;  y-x,  x,  ^-z;  y,  x-y,  ^-z; 

X,  x-y,  z+^;  y,  x,  z+|;  y-x,  y,  z-f-i 

Space-Group  Dsh. — (Hexagonal  Axes.) 

One  equivalent  position : 

(a)  0  0  0.  (b)  0  0 1. 

Two  equivalent  positions : 

(c)  HO;    fiO.  (e)  OOu;    0  0  u. 

K^J     3  1  2  >        3    3   2- 

Three  equivalent  positions : 

(f)   uuO;    OuO;    uOO.  (g)  uu|;    Ou|;    uO^ 

Four  equivalent  positions : 

(V,\     12,,.        2   1  fi  .        12,-,.        21,, 
W     3  I  U,        3    3  U,        3    3  U,        3    3  U. 

Six  equivalent  positions: 


(i)   u  u  V 

(j)  uvO 

vuO 

(k)  uvi 


vu| 


Ouv;    uOv;    tiOv;    uuv;    Ou^. 
v-u,  u,  0;    V,  u— V,  0; 
u,  v-u,  0;    u-v,  V,  0. 
v-u,  u,  I;    V,  u-v,  I; 
u,  v-u,  I;    u-v,  V,  |. 


Twelve  equivalent  positions: 

(1)    xyz;                y-x,  x,  z;    y,  x-y,  z; 

X,  y-x,  z;    yxz;  x-y,  y,  z; 

xyz;                y-x,  x,  z;    y,  x-y,  z; 

X,  y-x,  z;    yxz;  x-y,  y,  z. 

Space-Group  Dg^. — (Hexagonal  Axes.) 

Two  equivalent  positions: 

(a)  000;    OOi  (c)  Hi;    IH. 

(b)ooi;   oof.  (d)  HI;   Hi. 

Four  equivalent  positions: 

(e)  OOu;    OOti;    0,  0,  |-u;    0,  0,  u-|-|. 
(0  Hu;    Hu;    i  i  l-u;    |,  i  u-h|. 


160  THE  HEXAGONAL  SPACE-GROUPS  Da^-Ce. 

Space-Group  Da^^  (continued). 
Six  equivalent  positions: 

(g)  uuO;    OtiO;    uOO;    uu|;    Ou|;    u  0^ 
(h)  uv|;    v-u,  u,  i;    v,  u-v,  i; 
vuf;    u,  v-u,  f;    u-v,  v,  f. 

Twelve  equivalent  positions : 


xyz; 

y-x,  X,  z; 

y,  x-y. 

z; 

X,  y-x,  z; 

yxz; 

x-y,  y, 

z; 

X,  y,  i-z; 

y-x,  X,  h-z; 

y,  x-y. 

5-z; 

X,  y-x,  z+^; 

y,  X,  z+^; 

x-y,  y, 

z+i 

C.  HEXAGONAL  TETARTOHEDRY. 
Space-Group  CJ. — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  OOu. 
Two  equivalent  positions: 

(b)i!u;    flu. 
Three  equivalent  positions: 

(c)  Hu;    0|u;    fOu. 
Six  equivalent  positions : 

(d)  xyz;    y-x    x,  z;    y,  x-y,  z; 
xyz;    x-y,  x,  z;    y,  y-x,  z. 

Space-Group  Ce- — (Hexagonal  Axes.) 
Six  equivalent  positions : 

(a)  xyz;  y-x,  x,  z-j-^;    y,  x-y,  z-|-f; 

X,  y,  z+l;    x-y,  X,  z+f;    y,  y-x,  z+i 
Space-Group  Ce- — (Hexagonal  Axes.) 
Six  equivalent  positions: 

(a)  xyz;  y-x,  x,  z+f;    y,  x-y,  z+f; 

X,  y,  z+f;     x-y,  X,  z+f;     y,  y-x,  z+f. 

Space-Group  Ce. — (Hexagonal  Axes.) 

Three  equivalent  positions: 

(a)  OOu;    0,  0,  u+f;    0,  0,  u+f. 

(b)  ffu;    0,  f,  u+f;    f,  0,  u+f. 

Six  equivalent  positions: 

(c)  xyz;     y-x,  x,  z+f;     y,  x-y,  z+f; 
xyz;     x-y,  x,  z+f;     y,  y-x,  z+f. 


THE   HEXAGONAL   SPACE-GROUPS   Cfl-Coy.  161 

Space-Group  C*. — (Hexagonal  Axes.) 
Three  equivalent  positions: 

(a)  OOu;    0,  0,  u+h    0,  0,  u+f. 

(b)  Hu;    0,  h  u+h    h  0,  u+|. 

Six  equivalent  positions: 

(c)  xyz;  y-x,  x,  z+|;  y,  x-y,  z+f; 
xyz;    x-y,  x,  z-\-\;    y,  y-x,  z+f. 

Space-Group  Q%. — (Hexagonal  Axes.) 

Two  equivalent  positions : 

(a)  OOu;    0,  0,  u+|.  (b)  H  u;    f  i  u+i. 

Six  equivalent  positions : 

(c)  xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  y,  z+l;    x-y,  x,  z+|;    y,  y-x,  z+|. 

D.  HEMIMORPHIC  HEMIHEDRY. 
Space-Group  Cjy. — (Hexagonal  Axes.) 
One  equivalent  position: 

(a)  OOu. 
Two  equivalent  positions: 

(b)Hu;    fiu. 
Three  equivalent  positions: 

(c)  Hu;    OJu;    iOu. 
Six  equivalent  positions : 

(d)  uuv;    Ouv;    uOv;    utiv;    Ouv;    uOv. 

(e)  utiv;    2u,  %  v;    u,  2u,  v;    uuv;    2u,  u,  v;    u,  2u,  v. 

Twelve  equivalent  positions: 

(f)  xyz;  y-x,  x,  z;  y,  x-y,  z; 
xyz;  x-y,  x,  z;  y,  y-x,  z; 
X,  y-x,  z;  yxz;  x-y,  y,  z; 
X,  x-y,  z;    yxz;  y-x,  y,  z. 

Space-Group  Cgy. — (Hexagonal  Axes.) 

Two  equivalent  positions: 

(a)  OOu;    0,  0,  u-|-^. 

Four  equivalent  positions: 

(b)Hu;    f^u;    f,  i  u+^;    i  f,  u+i 


162  THE   HEXAGONAL   SPACE-GROUPS   Cey-Cfi. 

Space-Group  Cev  (continued). 
Six  equivalent  positions: 

(c)  Hu;  0|u;  ^Ou; 

h  h  u+l;    0,  i  u+l;    i  0,  u+i 

Twelve  equivalent  positions: 

(d)  xyz;  y-x,  x,  z;    y,  x-y,  z; 
xyz;  x-y,  X,  z;    y,  y-x,  z; 

x,  y-x,  z+l;    y,  x,  z+J;    x-y,  y,  z+^; 
X,  x-y,  z+^;    y,  x,  z+|;    y-x,  y,  z+i 

Space-Group  Cev- — (Hexagonal  Axes.) 

Two  equivalent  positions : 

(a)  OOu;    0,  0,  u+l. 

Four  equivalent  positions : 

(d)     3   3  U;        3,     3,    U-t-2j        3    3  U>       t>     3?    ^"r^* 

Six  equivalent  positions: 

(c)  uuv;  Otiv;  uOv; 

u,  u,  v+l;    0,  u,  v+l;    u,  0,  v+|. 

Twelve  equivalent  positions: 

(d)  xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  y,  z+l;  x-y,  X,  z+|;    y,  y-x,  z+^; 

X,  y-x,  z;  yxz;  x-y,  y,  z; 

X,  x-y,  z+l;     y,  x,  z+l;  y-x,  y,  z+^. 

Space-Group  CeV — (Hexagonal  Axes.) 

Two  equivalent  positions : 

(a)  OOu;    0,  0,  u+|.  (b)Hu;    I,  i  u+i 

Six  equivalent  positions: 

(c)  utiv;  2u,  u,  v;  u,  2u,  v; 

u,  u,  v-h^;    2u,  u,  v+?;    ti,  2u,  v+§. 

Twelve  equivalent  positions: 

(d)  xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  y,  z+l;        x-y,  X,  z+?;    y,  y-x,  z+l; 

X,  y-x,  z+J;    y,  x,  z-M;  x-y,  y,  z+|; 

X,  x-y,  z;  yxz;  y-x,  y,  z. 

E.  PARAMORPHIC  HEMIHEDRY. 

Space-Group  CeV — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  0  0  0.  (b)  0  0  i 


THE   HEXAGONAL   SPACE-GROUPS   Ceh-Dj.  163 


Space-Group  Ceh  (continued). 
Two  equivalent  positions : 


(c)  HO;    UO.  (e)  OOu;    OOu. 

K^J     3    3   2  >        3    3   2' 


Three  equivalent  positions: 

(f)  HO;   o|0;   100.  (g)  HI;   OH;   Hi 

Four  equivalent  positions: 

/'V.^     12,,.        21,,.        1    2  ,T  .        2   1  ,T 
W33U,        3    3  U,        33U,       ^lU. 

Six  equivalent  positions : 

iOu. 


(i)   Hu; 

Oiu; 

|0u 

Hu; 

0|u 

(j)   uvO; 

v-u, 

u,  0; 

V,  u-v, 

0; 

uvO; 

u-v, 

u,  0; 

V,  v-u. 

0. 

(k)uv|; 

v-u, 

u,  1; 

V,  u-v. 

1; 

Q^l; 

u-v, 

u,  h; 

V,  v-u, 

1 

2' 

Twelve  equivalent  positions: 

(1)  xyz;  y-x,  x,  z;  y,  x-y,  z; 
xyz;  x-y,  x,  z;  y,  y-x,  z; 
xyz;  y-x,  x,  z;  y,  x-y,  z; 
xyz;    x-y,  x,  z;    y,  y-x,  z. 

Space-Group  Cei. — (Hexagonal  Axes.) 


Two  equivalent  positions: 

(a)  0  0  0;    00  i 

(b)  0  0|;    oof. 

(c)  HO;    fH 

(d)HI;   HO 

Four  equivalent  positions: 

(e)  OOu;    OOu;    0,  0,  |-u;    0,  0,  u-f-i 
/f\    12,,.     12,-;.     2    1    1     ,, .     2    1    ,,  I  1 

W      3   3U,       35U,       5,    ^,    jr  — U,       3,     3,    Ui-2. 

Six  equivalent  positions: 

('c^lil-     n4i'     401-     111.     oil.     ini 
(h)  uvO;    v-u,  u,  0;    v,  u-v,  0; 
uv|;    u-v,  u,  I;    v,  v-u,  i 

Twelve  equivalent  positions: 

(i)    xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  y,  z-f-|;    x-y,  X,  z^-§;  y,  y-x,  z+|; 

xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  y,  l-z;    x-y,  x,  |-z;  y,  y-x,  |-z. 

F.  ENANTIOMORPHIC  HEMIHEDRY. 
Space-Group  DJ. — (Hexagonal  Axes.) 
One  equivalent  position: 

(a)  0  0  0.  (b)  0  0  h 


164  THE   HEXAGONAL    SPACE-GROUPS   dJ-d|. 

Space-Group  DJ  {continued). 
Two  equivalent  positions: 

(c)  HO;    UO.  (e)  OOu;    OOu. 

f/\\     12    1.        2    11 
\MJ    ■532;        332' 

Three  equivalent  positions: 

(f)  HO;   oio;   ioo.  (g)  Hi;   OH;   hoh 

Four  equivalent  positions: 

\M)     3   3^;       3    3  U;        3    3  U;        3   3  U. 

Six  equivalent  positions: 

(i)  Hu 
(J)  uuO 
(k)  uu| 


(1)   utiO 
(m)  u  u  I 


0|u;  iOu;  Hti;  OH;  |0u. 
OuO;  uOO;  uuO;  OuO;  u  0  0. 
Ou|;    GO  I;    uu|;    Ou|;    u  0  i. 


2u,  u,  0;    u,  2u,  0;    uuO;     2u,  u,  0;    u,  2u,  0. 
2u,  u,  I;    u,  2u,  I;    uu|;    2u,  u,  ^;    u,  2u,  |. 


Twelve  equivalent  positions : 

(n)  xyz;                y-x,  x,  z;    y,  x-y,  z; 

xyz;                x-y,  x,  z;    y,  y-x,  z; 

X,  y-x,  z;    yxz;  x-y,  y,  z; 

X,  x-y,  z;    yxz;  y-x,  y,  z. 

Space-Group  T>1. — (Hexagonal  Axes.) 

Six  equivalent  positions: 

(a)  OuO;    uO|;    tiuf;    Ou|;    uOf;    uu^. 

(b)  uura;    2u,  u,  f;    u,  2u,  Y2;    ^^¥2',    2u,  u,  i;    u,  2u,  rz- 

Twelve  equivalent  positions: 

(c)  xyz;  y-x,  x,  z+i;  y,  x-y,  z+|; 
X,  y,  z+i;  x-y,  x,  z+|;  y,  y-x,  z+i; 
X,  y-x,  z;  x-y,  y,  f-z;    y,  x,  |-z; 

X,  x-y,  l-z;    y-x,  y,  i-z;    y,  x,  f-z. 

Space-Group  1l>1. — (Hexagonal  Axes.) 
Six  equivalent  positions: 

(a)  OuO;    uOf;    uu^;    Ou|;    uO^;    uuf. 


(b)uura;    2u,  u,  f ;    u,  2u,  i^;    uufa;    2u,  u,  i;    u,  2u, 
Twelve  equivalent  positions: 

(c)  xyz;  y-x,  x,  z-|-f;  y,  x-y,  z+^; 

X,  y,  z+i;  x-y,  X,  z-f-^;  y,  y-x,  z-M; 

X,  y-x,  z;  x-y,  y,  ^-z;  y,  x,  f-z; 

X,  x-y,  i-z;  y-x,  y,  f-z;  y,  x,  ^-z. 


---   y 

12- 


THE   HEXAGONAL   SPACE-GROUPS   I>t-T>t 


165 


Space-Group  Dg. — (Hexagonal  Axes.) 
Three  equivalent  positions: 

(a)  0  0  0;    oof;    00  i 

(b)  00^;    OOi;    OOf. 

Six  equivalent  positions: 


(c)  ^  ^  3 ; 


0^0; 
nil. 


1  n  2 


(e)  OOu 
OOu 

(f)  Hu 
iOu 

(g)  uui 
(h)  uuf 
(i) 


uu| 


(J)  uuf 


0,  0,  u+f ;  0,  0,  u+i; 
0,  0,  l-u;  0,  0,  f-u. 
0,  h  u+f;  i  0,  u+f; 


0,  2}     3~Ui   ^>  ?>  a   ^• 


OuO;  uOf; 

Oui;  uOf; 

2u,  ti,  0;  u, 

2u,  u,  I;  u, 

Twelve  equivalent  positions: 
(k)  xyz;  y-x,  x, 

xyz;  x-y,  x, 

X,  y-x,  z;    x-y,  y, 


2 
3 

uuf; 
uuf; 


2u, 
2u, 


OuO; 

Ouf; 
uuf; 
uuf; 


uO 
uO 

2u, 

2u, 


0; 

1 . 


u,  2u, 
u,  2u, 


z+f; 
z+f; 

3  ^7 


X,  X-y,  z;    y-x,  y,  t-z; 
Space-Group  Dg. — (Hexagonal  Axes.) 
Three  equivalent  positions: 
(a)  0  0  0;    OOf;    OOf. 


(b)  OOf; 


y,  x-y,  z+f; 
y,  y-x,  z+f; 
y,  X,  f-z; 
y,  X,  f-z. 


/n\    111.     n  1  1. 

\^J     "5  2    6)        "  ^  3  > 

(d)  Iff;    OfO; 


i  n  1 

2^3' 


Six  equivalent  positions: 


(e)  OOu 
OOu 

(f)  ffu 


fOu 


(g)  uuf 
(h)  uuf 
(i)  uuf 


0,  0,  u+f;  0,  0,  u+f; 

0,  0,  f-u;  0,  0,  f-u. 

0,  f,  u+f;  f,  0,  u+f; 

0,  f,  f-u;  f,  f,  f-u. 

Ouf;  uOf;  uuf;  Ouf;  uOf. 

OuO;  uOf;  uuf;  OuO;  u  0  f . 

2u,  Q,  f;  u,  2u,  f;  uuf;  2u,  u,  f;  u,  2u,  f. 

2u,  u,  0;  u,  2u,  f;  uuf;  2u,  u,  0;  u.  2u,  f. 


(J)   uuf 
Twelve  equivalent  positions: 

(k)  xyz;  y-x,  x,  z+f;  y,  x-y,  z+f; 

xyz;  x-y,  x,  z+f;  y,  y-x,  z+f; 

X,  y-x,  z;    x-y,  y,  f-z;  y,  x,  f-z; 

X,  x-y,  z;     y-x,  y,  f-z;  y,  x,  f-z. 

Space-Group  DJ. — (Hexagonal  Axes.) 
Two  equivalent  positions: 


(a)  0  0  0; 
(b)OOf; 


(c) 
(d) 


12  1.        2   13 
3    3   4>        3    3   4- 


12    3.. 

3    3   4; 


2  11 

3  ^  ?• 


166 


THE   HEXAGONAL   SPACE-GROUPS   De-Dgh. 


Space-Group  Dg  (continued). 
Four  equivalent  positions: 


(e)  OOu;    OOu;    0,  0,  u+f;    0,  0,  |-u. 

^.1/      3    3   ">        3    3   ^)        3)     3j     l^   r2>        3>     3>     2        '^• 


Sto;  equivalent  positions : 

(g)  uuO;    OuO;    uOO;    uu^;    Ou^;    uOi 

(h)  uui;    2u,  u,  i;    u,  2u,  i;    uuf;    2u,  u,  f;    u,  2u,  i 

Twelve  equivalent  positions: 

(i)    xyz;  y-x,  x,  z;  y,  x-y,  z; 

X,  y,  z+l;  x-y,  x,  z+|;  y,  y-x,  z+|; 

X,  y-x,  z;  x-y,  y,  z;  yxz; 

X,  x-y,  |-z;  y-x,  y,  ^-z;  y,  x,  §-z. 

G.  HOLOHEDRY. 
Space-Group  Dgh. — (Hexagonal  Axes.) 
One  equivalent  position : 

(a)  0  0  0.  (b)  0  0 1. 

Two  equivalent  positions: 


(c)  HO; 
(d)P^ 


HO. 
IH. 


(e)  OOu;    OOu. 


1 1> 
Three  equivalent  positions: 

(f)   HO;    0|0;    fOO. 
Four  equivalent  positions: 


(g) 


111. 

^  "2  It 


OH; 


1  0  i 

2  '-'  2- 


(h) 


3    3  U,        3    3  U,       I   3  U,       ^  ^  U. 


Six  equivalent  positions: 

(i)   Hu;  Oiu;    §0u;    Hu; 

(j)  uuO;  OiiO;    uOO;    tiuO; 

(k)  uu|;  Otii;    uO|;    au^; 

(1)   uQO;  2u,  u,  0;    u,  2u,  0; 

(m)uu^;  2u,  u,  ^;    u,  2u,  |; 

Twelve  equivalent  positions : 


OH; 

iOu. 

OuO; 

uOO. 

Oui; 

uOi 

auO; 

2u,  u,  0; 

u,  2u, 

0 

Qui; 

2u,  u,  i; 

u,  2u, 

i. 

(n)  u  u  V 
uu  V 

(o)  uuv 
uu  V 

(p)  u  V  0 
uvO 


Ouv;    uOv;    tiuv;    Ouv;    uOv; 
Ouv;    tiOv;    uuv;    Ouv;    uOv. 
2u,  u,  v;    u,  2u,  v;    uuv;    2u,  u,  v;    u,  2u,  v; 
v;    uuv;    2u,  u,  v;    u,  2u,  v. 

V,  u-v,  0; 

V,  v-u,  0; 

vuO; 

vuO. 


2u,  ti,  v;  u,  2u 
V— u,  u,  0 
u— V,  u,  0 
Q,  V— u,  0;  u— V,  V,  0 
u,  u— V,  0;    V— u,  V,  0 


THE  HEXAGONAL  SPACE-GROUPS  Dei-Doh- 


167 


Space-Group  Dgh  (continued). 


v-u,  u,  I;  V,  u-v,  i; 

u-v,  u,  I;  V,  v-u,  ^; 

u,  v-u,  I;    u-v,  V,  h;  vu|; 

u,  u-v,  ^;    v-u,  V,  ^;  vu|. 


(q)  uv^; 


Twenty-four  equivalent  positions: 


(r)  xyz; 
xyz; 

X,  y-x,  z; 
X,  x-y,  z; 
xyz; 
xyz; 
X,  y-x,  z: 


y-x,  X,  z;  y,  x-y,  z; 

x-y,  X,  z;  y,  y-x,  z; 

x-y,  y,  z;  yxz; 

y-x,  y,  z;  yxz; 

y-x,  X,  z;  y,  x-y,  z; 

x-y,  X,  z;  y,  y-x,  z; 

x-y,  y,  z;  yxz; 


X,  x-y,  z;    y-x,  y,  z;    yxz. 
Space-Group  Deh- — (Hexagonal  Axes.) 
Two  equivalent  positions: 

(a)  000;    OOi  (b)  OOJ;    OOf. 

Four  equivalent  positions: 

HI. 
Hi;   HI;   Hi 

(e)  OOu;    OOQ;    0,  0,  |-u;    0,  0,  u+i 
/Six  equivalent  positions: 


(c)    H  Oj      f  I  Oj       3  3  5? 

(d)  i  !  i; 


1 1 . 

113. 

1  1  J) 


11) 
1    3  . 

1  4, 


(f)  HO;    0^0;    100;    | 

(g)  Hi;   OH;   Hi; 

Etgf^f  equivalent  positions: 

(h)  Hu;    fiu;    f,  i  u+i;    i  f,  u+i; 

Hu;   HQ;   !,  i  I-u;   i  !,  i-u. 

Twelve  equivalent  positions: 
(i)   Hu;    Oiu;    iOu; 
Hii;    Oiu;    iOO; 
i  I,  i-u;    0,  i,  i-u;    i  0,  i-u; 
i  i  u+i;    0,  i,  u+i;    i,  0,  u+i. 

OuO;    tiOO;    uuO;    OuO;    uOO; 

Oui;    uOi;    uui;    Oui;    uOi. 

2u,  u,  0;    u,  2u,  0;    tiuO;    2u,  u,  0;    u,  2u,  0; 

2u,  u,  i;    u,  2u,  i;    uui;    2u,  u,  i;    ti,  2u,  i. 

v-u,  u,  i;    V,  u-v,  i; 

u-v,  V,  i;    V,  v-u,  i; 

Q,  v-u,  I;    u-v,  V,  f; 
vui;    u,  u-v,  f;    v-u,  v,  |. 


(j) 

uuO 

uui 

(k) 

uuO 

uui 

(1) 

u  v| 

u  vi 

vuf 

168 


THE   HEXAGONAL  SPACE-GROUPS  Da^-De^ . 


■z; 


Space-Group  Del  {continued). 

Twenty-four  equivalent  positions: 
(m)  xyz; 
xyz; 

X,  y-x,  z; 
X,  x-y,  z; 
X,  y,  |-z; 
X,  y,  |-z; 
X,  y-x,  z+l; 
X,  x-y,  z+i; 

Space-Group  Dgh. — (Hexagonal  Axes.) 
Two  equivalent  positions: 

(a)  0  0  0;    OOi  (b)  OOJ;    0  0  f . 

Four  equivalent  positions : 
(c)  IfO; 

(d)Hi; 

(e)  OOu;    OOii;    0,  0,  |-u;    0,  0,  u+|. 
Six  equivalent  positions: 


y-x,  x,  z; 
x-y,  x,  z; 
x-y,  y,  z; 
y-x,  y,  z; 
y-x,  X,  i- 
x-y,  X,  ^-z; 
x-y,  y,  z+i; 
y-x,  y,  z+l; 


y,  x-y,  z; 

y,  y-x,  z; 

yxz; 

yxz; 

y,  x-y,  §-z; 

y,  y-x,  h-z; 
y,  X,  z-fl; 
y,  X,  z+^. 


111. 

3   3   2, 

2  13. 

3  3   4, 


2  1  f). 

3  3*-', 

2  11. 

3  3   4, 


111 
3  3  2- 
ill 
3    3    4' 


/f\     111.      nil-      inl'      111-      04^'      i-Ol 
W     15  4,       "¥4,        2^4,       ¥¥4,       "5  4,       t  "  4- 

(g)  uuO;    OuO;    uOO;    Qui;    Ou^;    uO|. 
Etgf^f  equivalent  positions: 

(h)  Hu;    Hu;    f ,  i  u+l;    i  i  u-|-|; 


i 


^  U,       53U,       S,    ^,    5  — U,       3,    S,    ^       U. 


Twelve  equivalent  positions: 


(i) 

uu  i 

uuf 

(J) 

uvO 

u  v| 

vuO 

vG| 

(k) 

UU  V 

UUV 

2u,  u,  i;  u,  2u,  i;  uuf;  2u,  u,  f; 
2u,  u,  f;  u,  2u,  f;  uui;  2u,  u,  i; 
V— u,  u,  0;  V,  u— V,  0; 
V,  u-v,  i; 
u-v,  V,  0; 
v-u,  V,  J. 
uOv; 
QOv; 


u-v,  u,  i; 

u,  V— u,  0; 

u,  u-v,  i; 

Oflv; 

Oflv; 

%  u,  v-l-i;    0,  u,  v+^;    u,  0,  v-f-^; 
G,  G,  i-v;    0,  u,  i-v;    u,  0,  |-v. 
Twenty-four  equivalent  positions: 


(1)   xyz; 

X,  y,  z-M; 

X,  y-x,  z; 

X,  x-y,  i-z; 

xyz; 

X,  y,  i-z; 

X,  y-x,  z; 

X,  x-y,  z-l-l; 


y-x,  X,  z;         y,  x-y,  z; 
x-y,  X,  z+i;    y,  y-x,  z+i; 


x-y,  y,  z; 
y-x,  y,  i-z; 
y-x,  X,  z; 
x-y,  X,  i-z; 
x-y,  y,  z; 
y-x,  y,  z+l; 


yxz; 

y,  X,  i-z; 

y,  x-y,  z; 

y,  y-x,  i-z; 

yxz; 

y,  X,  z-\-^. 


G,  2u,  f ; 
G,  2G.  I 


THE   HEXAGONAL   SPACE-GROUP   DeV  • 


169 


Space-Group  D6*h. — (Hexagonal  Axes.) 
Two  equivalent  positions : 


(a)  0  0  0;    00  i 

(b)  OOi;    OOf. 

Four  equivalent  positions : 


V^/     3    3   4  )        3    3    4- 


(d)  I 


12    3. 
3    3   4; 


111 
3    3   4- 


(e)  OOu;    OOti;    0,  0,  ^-u;    0,  0,  u+f. 
(0    H 


2     1     n_|_l  •      2  1,-,. 
3j     3>    "   r2>        3    3"? 


12       1 

3j     3>    ^ 


-u. 


Six  equivalent  positions : 

(g)  HO;   010;   ^00;   H^;   OH;   hoh 

(h)  uui;    2u,  u,  i;    u,  2u,  i;    uuf;    2u,  u,  |;    u,  2u,  f. 
Twelve  equivalent  positions : 


(i)    uuO 


uu^ 


(j)   uvi 


ti  v| 
vuf 
V  ti  J 


OuO;    uOO;    uuf;    Ou|;    uO|; 
Oti^;    uO^;    uuO;    OuO;    uOO. 


v-u,  u,  t 

3  . 


V,  u-v,  I; 


u-v,  u,  t;    V,  v-u,  i; 
u,  V 


u,  f;    u-v,  V,  f; 


u,  u-v,  i;    v-u,  V,  i. 


(k)  uuv;  2u,  u,  v;  u,  2u,  v; 

uuv;  2u,  u,  v;  u,  2u,  v; 

u,  ti,  ^-v;  2u,  u,  §-v;  u,  2u,  |-v; 

u,  u,  v+l;  2u,  u,  v+h;  %  2u,  v+^. 


Twenty-four  equivalent  positions : 


(I)    xyz; 

X,  y,  z+^; 
X,  y-x,  z; 
X,  x-y,  \-z] 
X,  y,  §-z; 
xyz; 


y-x,  x,  z; 


y,  x-y,  z; 


x-y,  X,  z-\-\;    y,  y-x,  z-f-^; 
yxz; 


x-y,  y,  z; 

y-x,  y,  h- 

y-x 

x-y,  x,  z; 


z; 


X,  2~z; 


y,  X,  t-z; 

y,  x-y,  \-z] 
Y,  y-x,  z; 


X,  y-x,  z+i;    x-y,  y,  z-|-|;    y,  x,  z-H; 


X,  x-y,  z; 


y-x,  y,  z; 


yxz. 


170 


tables:  triclinic  and  monoclinic  systems. 
TABLES. 


The  following  tables  provide  a  summary  of  the  number  and  kinds  of  the 
different  arrangements  to  be  obtained  from  each  of  the  space-groups.  In 
these  tabulations  the  symbol  1  (0),  for  instance,  signifies  one  arrangement 
(a  special  case)  having  no  variable  parameters;  similarly  the  symbol  3  (2) 
would  mean  three  arrangements  having  two  variable  parameters  each. 

Table  3.— TRICLINIC  SYSTEM. 


Space-Group. 

Number  of 
equivalent  positions. 

1 

2 

A.  Hemihedry: 
Ci 

1(3) 
8(0) 

1(3) 

B.  Holohedry: 
ci 

Table  4.— MONOCLINIC  SYSTEM. 


Space-Group. 

Number  of  equivalent  positions. 

1 

2 

4 

8 

A.  Hemihedry: 

CI 

2(2) 

4(1) 
8(0) 

1(3) 
1(3) 
1(2) 

1(3) 
1(3) 
2(1) 

4(1);  2  (2) 
4  (0);  1  (2) 

4(0) 

4(0);  2(1) 
4(0) 

1(3) 
1(3) 

1(3) 

1(3) 

1(3) 
f2(0) 

2(1) 
ll(2) 

1(3) 

1(3) 
4  (0);  1  (1) 

CI 

c^ 

ct 

B.  Hemimorphic 
hemihedry: 
d 

c^ 

c; 

C.  Holohedry: 

CL 

1  (3) 

CL 

Cjh 

CSh 

l'(S) 

tables:  orthorhombic  system. 


171 


.2 

■s 
•s 

J 

s 

i 

C5   CO       .       .       . 

. 

00 

c? 

o 

"* 

cv?  CO  CO  CO  CO  cv?  J^  CO  c^  CO  "-^  ??  3  S ""' ""*  d  d    •^'-^'~' 

1— li— li-Hi— IrHi-HrHi-Hi-Hj— 1   ^  i— 1    CO    CO   ^^  t— 1    i-H        •    CI    C<J   ^ 

w 
o 

B 

N 

rH 

PQ 

•41 

E-I 

■^ 

c 
c 

1 

c 
ei 
C 

a 

J 
> 

5 

> 
I 

^     '''.'.'.'.'.'.'....■..'.'.'.'.'.'.      '. 

<» ■                     ■     ■. 

"c^a 

i; 

^ 

.^    ''''.'.'.'.'.'.'.'.'.'.'.'.'.'.'.'.   '.   :. 

J    •    •    :    :    :    : -.:::;: 

r«ft 

.a ■                                   ..... 

j|00000U00000Q0Q0U0000U0 

172 


tables:  orthorhombic  system. 


«*5 


O 


o 


S3 

"3 


a 


00 


-«*< 


CO  CO  tH  CO  CO 
1-1    1-4    O    r-l    rH 


COCOCOCOCOCOCOCOCOCO         CO         cocococo 


COCOCOCOi— ^i-hOi— ii— • 
,_(_<r-(rHC^t^Tt<C>CO 


'I  (N   1-1  (M  !-<  (N   --I  IM   1-1^ 
■«0  1— I  CO  CO  (N  t-l  CO  (N 


"®  O*  ^  O' Ch"  o' Ch*  o' Ch' o','^^  '^  '~',Ch  o'o'<^ 
(MO0C<J(M(NCO(N<N(N  (M(M(>J 


.  O      .  O, 


rH    0,30, 

(M  -^  00  -ti 


O 

00 


QO 


tables:  orthorhombic  system. 


173 


fsi 

[SSw 

PO 

•    1-H    i-H 

^^.-. 

(M  ^ 

^^ 

s                     ^-^ 

CO   CC  CO   CC  '^  '^   CO 

■v                     ^^"^   ^•"•^ 

vO 

M 

M 

",         ^ 

CO  CO 

i-H 

T-l 

^   ^   ^   ^  ^g^ 
CO  <M 

1— 1 

rH    r-4 

.1 

^_^ 

^ ^^ ^^ ^^ ^,— ,         , 

^ ^,_^ 

a 

(M  C^???^— 1          C^ 

»-i  <N 

» 

+J 

N*-^  >*-^ 

t3 

a 

00 

s 

'S5 

^C^O'^ 

-^■*.-i(N->*C03-C0 

sE 

'S'll'^. 

1— ( 

I— 1    i-H 

(M    r-l    (M 

'-' ^■^i^o'o''^  O 

-,-H    -* 

—  b3 

c 

O 

p2 
o3 

,-(  lO  iQ  IM  '^          T-H 

(N  (N 

_> 

:^ 

'3 

^— . 

\            <^* 

^-"^                .'■"V 

^-^s 

o* 

(N 

>-H 

T-H                   1-^ 

T-H 

0) 

^           >— » 

■S—^                N..^ 

**— ^ 

*© 

l-H 

T-H 

o 

;     "=° S •"'  o  o    :5 

■    0 

■»                   •     ^H 

c» 

Tj< 

■^                   .      •  *^ 

CO 

J 

O 

^     2 

(M 

g-O  3-0^  (N       •  «C 

■>* 

■  s 

O 

a 

(N 

(N 

c^       0 

■* 

s 

s 

o 

„ ^ ,— 

^ 

w 

M 

0 5 

, 

P5 

■<*< -^ 

o 

w 

H 

S 

? 

1 

>-^ 

H 

.J 

n 

£5 

C 

u 

c 

*. 

C 

s 

i 

g 

1. 

a: 

i 

j^ 

flO 

»         0         ^         C4         <0         -W         lA 

<e 

-■-■     1 

3           •-«  - 

a        -.j3«j:o»jc»xc-ij3«jaNj 

a        « J 

a          Mjsr<d 

> 

> 

> 

> 

> 

> 

> 

> 

> 

> 

> 

> 

> 

174 


tables:  tetragonal  system. 

Table  6.— TETRAGONAL  SYSTEM. 


Space-Group. 


Number  of  equivalent  positions. 


16 


32 


A .  Tetartohedry  of  the 
second  Sort: 

S] 

SJ 


4(0) 


B.  Hemihedry  of  the 
i>econd  Sort: 
V^ 


4(0) 


Yl 
Yl 

yi 

VI. 

yi 
v^ 

Vi» 

vr 


3(1) 
4(0) 


2(0);  2(1) 

6(0) 
2  (0);  1  (1) 

2(0) 

4(0) 


C.  Tetartohedry: 

CI 

CI 

CI 

c\ 

c\ 

c: 


2(1) 


D.  Paramorphic 
hemihedry: 

Cih 

Cih 

Cib 

G4h 


4(0) 


2(0) 

1(1) 
3(1) 
l'(l) 


2(0);  2(1) 

6(0) 
2  (0);  1  (1) 

2(0) 

2(0) 


1(3) 
2(1) 


(1);  1 
7(1) 

(1);  1 
2(1) 
3(1) 
4(0) 
4(0) 
4(0) 
4(0) 


2(0);  1(1) 
2(0) 


1(3) 


1(3) 
1(3) 
1(3) 
1(3) 

(1);  2 
5(1) 
4(1) 
4(1) 
2(1) 
4(0) 

(1);  1 
2(1) 


(3) 
(3) 
(3) 
(3) 
(1) 
(1) 


1(1);  2(2) 
3(1);  1(2) 
2(0);  1(1) 
2(0);  2(1) 

2(0);  1(1) 

2(0) 


(2) 


(2) 


1(3) 
1(3) 
1(3) 
1(3) 

(1);  1 

4(1) 
1(3) 
1(3) 


(2) 


(3) 
(3) 


(3) 
(3) 


1(3) 
1(3) 


tables:  tetragonal  system. 

Table  6.— TETRAGONAL  SYSTEM  (Continued). 


175 


Space-Group. 


E.  HemiTnorphic 
hemihedry: 

a 


a. 

V^4v. 

CIO 
4v 
pll 

C12 
4v 


Number  of  equivalent  positions. 


2(1) 


F.  Enantiomorphic 
hemihedry: 

Dl 

D^ 

DJ 

DJ 

DJ 

d: 

D^ 

DJ 

d: 

Dl" 


G.  Holohedry: 

Dl,. 

DL 

DL 

DJh 

■L'4h 

DJ, 

dIh 

D8 
4h 

D» 
4b 


Dl 


A^4h. 

■L'4h- 

■L'4h- 

D18 
4h- 

■'-'4h- 

^4h- 


4(0) 


4(0) 


1(1) 
2(1) 
2(1) 
1(1) 
2(1) 
1(1) 
3(1) 

1(1) 


2(0);  2(1) 
2  (0);  1  (1) 


6(0) 
2(0) 


2(0) 


(0);  2 
4(0) 
4(0) 
2(0) 
4(0) 
2(0) 

(0);  1 

6(0) 
4(0) 

2'(0) 

2(0) 
2(0) 

2(0) 


(1) 


(1) 


3(2) 

1(2) 

(1);  1 

(1);  1 

1(1) 
1(1) 

2(2) 
2(1) 
1(1) 
2(1) 
1(1) 


7 
3 
3 
1 
9 
4 
3 
1 
2(0) 
2 


7 

(0) 
(0) 
(0) 

4 

(0) 
(0) 
(0) 

7 
(0) 

4 
(0) 

4 
(0) 

2 

(0) 
(0) 

4 

2 


1) 
1) 
1) 
1) 
1) 
1) 
1) 
1) 
1(1) 
0) 


1) 
2(1) 
2(1) 
1(1) 

1) 
1(1) 
1(1) 
1(1) 

1) 


(1) 

(1) 

(1) 

(1) 

1(1) 

0) 

0) 


1(3) 
(3) 
(3) 
(3) 
(3) 
(3) 
(3) 
(3) 
(2) 
(2) 
(2) 
(1) 


1(3) 
1(3) 


(3) 
(3) 
(3) 
(3) 
(3) 
(3) 
(1) 
(1) 


5(2) 
(1);  1 
(1);  1 

(0);4 

3(2) 


2(1); 
2(1); 
1(0); 
1(1); 
3(1); 
1(0); 
5  (1); 
3(1); 
1(1); 


(2) 
(2) 
(1) 

(2) 

(2) 

(1) 

(2) 
(2) 

(1) 

(2) 
(2) 
(2) 


(0);  1(1);  1(2) 
3(1);  1(2) 
1(0);  4(1) 
1(0);  3(1) 
2(0);  1(1) 
2(0) 


16 


1(3) 
1(3) 
1(3) 
1(3) 


(3) 
(3) 


(3) 

(3) 

(3) 

(3) 

(3) 

(3) 

(3) 

(3) 

(3) 
1(3) 
1(3) 
1(3) 
1(3) 
1(3) 
1(3) 
1  (3) 

1(1);  3  (2) 
2(1);  2(2) 
2  (1);  1  (2) 
1(0);  3(1) 


32 


1(3) 
1(3) 
1(3) 
1(3) 


176 


tables:  cubic  system. 


On 


0\ 


.  CC       .       .00 


s 


00 


i-H  CO       .00 


CO 

O 
I— I 


a 


o 

S 


O 


<M 


00 


VO 


.   1— I   CO       .CO 

•    Ol    T-H         -1—1 


CO       .  1— I  CO 


w   ■  2- 


O 


CO  CO  .  (N  00 


2-C 


1— (  1-H  O  O  O  i— I  o 

?— I     1— I     T— I     CI     1— (     I— I     C<J 


O 


o 


CO  I— I         £2-  "^  ^^ 
.-H   (N  ■^ '-I   (N   1-H 


i-H    1— I    (M 


(M       •  1-1  CO 


I— I    Ttl 


O 


o 


o  o 


ex 

I 


03 

CO 


^^ 


^H^HHH 


11 

%^    •  c>»  '^  J3  e^  j3  <•  X  *«■  j3  ><»  JS  «o  o:  1^  .C 

a  gHHHHHHH 


•8 


HHHHEhH 


cq 


tables:  cubic  system. 


177 


1— t 

.       .       .       .  CO  CO  CO  CO       .       . 
•    I-H    1-H    I-H    I-H 

ss 

s 

:   :gg;  :   :   :   : 

'.'.'.'.  C^^^^CO  CO 

•  1— *  1— ( 

1-H    I-H 

s 

(1)1 
(1)1 

s 

00 

;   ; ^^cc   ;   ;  n^ 

^-V  .^V  ^-V  ^-V  ^— V  ..— V-V  ^-V  C^     ^ 

COCOCOCOi-Hi-Hi— lO          r-< 

■  CO  M  1—1              •  1— 1 

i-Hi-Hl-Hi-HCO(Ml— li-HiZn^C^ 

1 

a 
12; 

1-H 

CN| 

!     !  i-H  ^H     i          i     ! 

•  •  •  •  3  •  3§  •  3 

<*i 

•  1— t  1— ( 

■    I-H         •    I-H    C^         -I-H 

T— ( 

<M  (M  i-i 

o 

CO  CO  ""^     !  -— t  CO  CO  1-H 

I-H    T— (    (3         •    C^    1—1    I-H    CO 
I-H 

S3TlTlTls  '■  •3b 

CO  oa  Ch"  ^  S  "^J     •     •  (N  c^ 

i-l    C^    1-H 

o 
O 

i 

2(0) 

1(1) 
1(1) 

•  33  •  •  •  §§32 

•    1-H     I-H                               .    (M     ^     ^    Cs^ 

3 

T-H               1-H                                               1-H 

t-4 

33  •  •^.33s 

C^    lO                •  3  '-H    1-H   (N 

' — ^  1-H   ' — ^  I-H        -        •        •        •    I-H        • 
1— '             1— <                 .... 

CO  S'^  s  ■   ■   ■   ■  s  ■ 

R 

1-H 

I-H            1-H                                       1-H 

00 

^ciro'o^o'333 

3§§3§^^  iS  ! 

9 

^Hi-H^HM^H^Hi-HOfl 

,-i    ^    ,-,    r-,    -^    C^    C<i         -I-H         • 

T 

«o 

32^   •    •  2   • 

^O    0,0         .         .         .         .    O,        . 

M 

1^  CO       •       •  I-H       •       •       • 

<N    1-H    CO    I-H         •         •         •         •    r-l         • 

i 

^ 

: So    :   : So   : 

:       :  SS  :   :   :   ;   : 

•  <N  (M       •       •  (N  IM       • 

•     •     •  (N  (N 

»*) 

S  ::::::   ; 
(M 

S  :;::.:::   : 
M :".■.; 

C4 

:  S  :   :  o   :   :   : 

•  SSS  •■■•§• 

•    rH         •         •    —1         •         • 

•    fH    i-H    1— 1                                   -I-H 

»-« 

s 

::::::        S 

C 

1 

U 

) 
i 
i 

! 

D.  Enantiomorphic 
hemihedry: 
o» 

C 

666666 

-2--. 
t?^ 

"c 

oc 

oooooo 

1^ 

178 


tables:  hexagonal  system. 


Table  8.— HEXAGONAL  SYSTEM. 
I.  Rhombohedral  Division 


Space-Group. 


A.  Tetartohelry: 

CI 

C^ 

CI 

Ci 


B.  Paramorphic 

hemihedry: 
CJi 

Csi 


C.  Hemimorphic 
hemihedry: 

CI 

Cjv 


Csv 

Cjv 
C.v 


D.  Enantiomorphic 
hemihedry: 
Dl 

D^ 

Df 

D^ 

T>1 

D'a 

DI 


E.  Holohedry: 
DL 

DL 

Dir...... 

DJa 

DL, 


Number  of  equivalent  positions. 


3(1) 


1(1) 


2(0) 
2(0) 


3(1) 
1(1) 


1(1) 


6(0) 
2(0) 


2(0) 

2(0) 
2(0) 
2(0) 


2(1) 
1(1) 


1(3) 
1(3) 
1(3) 
1(3) 


2(0) 
2(0) 


1(2) 

1(1) 

1(2) 

3(1) 

2(1) 

1(2) 

1(1) 

3(1) 

2(1) 

2(1) 

2(1) 

2(1) 

2(1) 

2(1) 

2(1) 

1(1) 

2(1) 

2(0);  1(1) 

2(0) 

4(0) 

2(1) 

2(0) 

2(0) 

1(1) 

2(0) 

2(0) 

1(1) 

2(1) 
2  0) 

I'd) 


(3) 
(3) 


(3) 
(3) 
(3) 
(3) 
(3) 
(3) 


(3) 
(3) 
(3) 
(3) 
(3) 
(3) 
(3) 


12 


2(1); 
1(0); 
2(1); 
1(0); 
2(1); 
1(0); 


1(2) 
1(1) 
1(2) 
1(1) 
1(2) 
1(1) 


1(3) 
1(3) 
1(3) 
1(3) 
1(3) 
1(3) 


tables:  hexagonal  system. 


179 


o 
O 

si 


tJ 

S 

-< 

o 

^ 

o 

o 

^ 

o 

» 

<  K 

u 

1—4 

T 

1 

05 

» 

»:3 

n 

< 

H 

a 


B 

3 


00 


CO  CO  CC  CO 


CO  ■^co  "^ 


CO  CO  CO  CO  CO  CO 


r-4    O 


o 

O 

.  o 

CC 

ec 

•  (M       • 

•55 

-SAg^ 

"e 

?> 

s  -« 

b   S   <» 

o 

<» 

§j  s  -«    ^ 

=         Oj-s;  ^i 

1 »)  m  -5  m  •»  S 

.  Tn 

para 
hemi 

■    £ 

-^0 

QQ 

Q 

ft ; ; ; ; ; ; 

s  « 

l*-2    :    :    :    :    :    : 


6 


05 


180 


tables:  hexagonal  system. 


• 

CO   CO   CO  CO 

1-H    1-H    1-H    I-H 

c^^c^c^ 

<VI 

CO  fC  fC  M~ 

CO  cv? 

CO  CO  CO  CO  CO  CO 

^»-H  (M  (M 

1-H    t— 1    »— 1    F- 

i-H    »— t 

T}^^^^ 

CO    r-H    I-H 

00 

.    1-H    1-H 
•    I-H    1-H 

I 

S^ 

I-H    I-H 

d 

m'^m"?^ 

~ 

IM  1- 

Ss"^ '-' 

a 

o 

(N    "-I    r-H    r- 

- 

I-H    r- 

lO  (N  (N  «0  «0  (M 

^  c^  o  o 

I-H    r-i 

> 
V 

o 

■^ 

i    1-H    1— 1 

s= 

I-H     !     !     !     !  1-H 

1-H    1-H 

s 

•     T— 1     .—1 

rH    <N 

rH        •         •        •        •    (N 

(M  (N 

»o 

■—I 

o 

s  •  -ss  • 

o   ;   :   : 

1-H 

(N 

(N       •      ■  M<  M<      • 

Cv|       .       .       . 

I-H 

1-H 

1-H 

N 

1— 1    1— 1    »— 1    i-H 

Tj 

• 

71  •  •      -s 

"^  o  Sg 

1-H    1— 1    I— 1    04 

g  .  .  .  .^ 

(M 

*H 

y~-> 

o 

■• 

t ; ; ;  : : 

g  :   :  : 

1-H 

(N 

■<5y 
'■'■* 

N 

(N      •      •      • 

6 

1 

CO 

5 

lie 

°c 

°c 

°c 

^  1 

-1 

'c 

1 

-1  :   :   :   :   :   : 

IQQQQQQ 

:     . 

t     . 

'o 

1 

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